2)^ 


U> 


IN  MEMORIAM 
FLORIAN  CAJORl 


1 


ELEMENTS   OF   ANALYTIC   GEOMETRY 


A  SERIES  OF  MATHEMATICAL  TEXTS 

EDITED    BY 

EARLE  RAYMOND   HEDRICK 


THE   CALCULUS 

By    Ellery    Williams    Davis    and    William    Charles 
Brenke. 
ANALYTIC   GEOMETRY  AND   ALGEBRA 

By  Alexander  Ziwet  and  Louis  Allen  Hopkins. 
ELEMENTS  OF  ANALYTIC   GEOMETRY 

By  Alexander  Ziwet  and  Louis  Allen  Hopkins. 
PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
COMPLETE   TABLES 

By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 
PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
BRIEF  TABLES 

By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 
ELEMENTARY   MATHEMATICAL   ANALYSIS 

By  John  Wesley  Young  and  Frank  Merritt  Morgan. 
PLANE    TRIGONOMETRY    FOR    SCHOOLS    AND    COL- 
LEGES 

By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 
THE   MACMILLAN  TABLES 

Prepared  under  the  direction  of  Earle  Raymond  Hedrick. 
PLANE   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 
PLANE   AND   SOLID   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 
SOLID  GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 
CONSTRUCTIVE   GEOMETRY 

Prepared  under  the  direction  of  Earle  Raymond  Hedrick. 


ELEMENTS 

OF 

ANALYTIC   GEOMETRY 


BY 

ALEXANDER  ZIWET 

PROFESSOR   OF    MATHEMATICS,    THE    UNIVERSITY   OF   MICHIGAN 
AND 

LOUIS  ALLEN  HOPKINS 

INSTRUCTOR   IN   MATHEMATICS,    THE    UNIVERSITY    OF   MICHIGAN 


THE   MACMILLAN   COMPANY 
1916 

All  rights  reserved 


COPTBIGHT,  1916, 

bt  the  macmillan  company. 


Set  up  and  electrotyped.     Published  October,  1916. 


Norinooti  ^ress 

J.  8.  Gushing  Co.  —  Berwick  &  Smith  C!o. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

As  in  most  colleges  the  course  in  analytic  geometry  is  pre- 
ceded by  a  course  in  advanced  algebra,  it  appeared  desirable 
to  publish  separately  those  parts  of  our  "  Analytic  Geometry 
and  Principles  of  Algebra  "  which  deal  with  analytic  geometry, 
omitting  the  sections  on  algebra.  This  is  done  in  the  present 
work. 

In  plane  analytic  geometry,  the  idea  of  function  is  intro- 
duced as  early  as  possible ;  and  curves  of  the  tovm  y  =f{x), 
where  f{x)  is  a  simple  polynomial,  are  discussed  even  before 
the  conic  sections  are  treated  systematically.  This  makes  it 
possible  to  introduce  the  idea  of  the  derivative ;  but  the  sec- 
tions dealing  with  the  derivative  may  be  omitted. 

In  the  chapters  on  the  conic  sections  only  the  most  essential 
properties  of  these  curves  are  given  in  the  text ;  thus,  poles 
and  polars  are  discussed  only  in  connection  with  the  circle. 

The  treatment  of  solid  analytic  geometry  follows  more  the 
usual  lines.  But,  in  view  of  the  application  to  mechanics,  the 
idea  of  the  vector  is  given  some  prominence ;  and  the  repre- 
sentation of  a  function  of  two  variables  by  contour  lines  as 
well  as  by  a  surface  in  space  is  explained  and  illustrated  by 
practical  examples. 

The  exercises  have  been  selected  with  great  care  in  order 
not  only  to  furnish  sufficient  material  for  practice  in  algebraic 
work,  but  also  to  stimulate  independent  thinking  and  to  point 


Hrpf\g^gr*g^M 


vi  PREFACE 

out  the  applications  of  the  theory  to  concrete  problems.  The 
number  of  exercises  is  sufficient  to  allow  the  instructor  to  make 
a  choice. 

ALEXANDER   ZIWET. 
LOUIS  A.    HOPKINS. 


CONTENTS 
PLANE  ANALYTIC   GEOMETRY 


PAGES 


Chapter  I.    Coordinates 1-21 

Chapter  II.    The  Straight  Line 22-36 

Chapter  III.    Relations  between  Two  or  More  Lines  .        .  37-50 

Chapter  IV.    The  Circle         . 51-70 

Chapter  V.    Polynomials 71-92 

Parti.       Quadratic  Function  —  Parabola     .         .         .  71-81 

PartIL     Polynomials 82-92 

Chapter  VI.    The  Parabola 93-115 

Chapter  VII.    Ellipse  and  Hyperbola 116-139 

Chapter  VIII.    Conic  Sections  —  Equation  of  Second  Degree  140-162 

Part  I.       Definition  and  Classification  ....  140-147 

Part  II.     Reduction  of  General  Equation      .         .         .  148-162 

Chapter  IX.    Higher  Plane  Curves 163-188 

Part  L       Algebraic  Curves 163-168 

PartIL     Special  Curves 169-177 

Part  III.   Empirical  Equations 178-188 

SOLID   ANALYTIC   GEOMETRY 

Chapter  X.    Coordinates 189-203 

Chapter  XL    The  Plane  and  the  Straight  Line   •        .        .  204-228 

PartL       The  Plane 204-218 

Part  II.     The  Straight  Line 219-228 

Chapter  XII.    The  Sphere 229-241 

Chapter  XIII.    Quadric  Surfaces  —  Other  Surfaces     .        .  242-265 

Tables 266-272 

Answers 273-276 

Index 277-280 

vii 


PLANE  AN^ALYTIC  GEOMETRY 

CHAPTER   I 
COORDINATES 

1.  Location  of  a  Point  on  a  Line.  The  position  of  a  point 
P  (Fig.  1)  on  a  line  is  fully  determined  by  its  distance  OP 
from  a  fixed  point  O  on  the  line,  if  we  know  on  which  side  of 
0  the  point  P  is  situated  (to  the  right  or  to  the  left  of  O  in 
Fig.  1).     Let  us  agree,  for  instance,  to  count  distances  to  the 


4-H^ 


t 


Fig.  1 


right  of  0  as  positive,  and  distances  to  the  left  of  0  as  negative ; 
this  is  indicated  in  Fig.  1  by  the  arrowhead  which  marks  the 
positive  sense   of  the  line. 

The  fixed  point  0  is  called  the  origin.  The  distance  OP, 
taken  with  the  sign  +  if  P  lies,  let  us  say,  on  .the  right,  and 
with  the  sign  —  when  P  lies  on  the  opposite  side,  is  called 
the  abscissa  of  P. 

It  is  assumed  that  the  unit  in  which  the  distances  are 
measured  (inches,  feet,  miles,  etc.)  is  known.  On  a  geographi- 
cal map,  or  on  a  plan  of  a  lot  or  building,  this  unit  is  indicated 
by  the  scale.  In  Fig.  1,  the  unit  of  measure  is  one  inch,  the 
abscissa  of  P  is  +  2,  that  of  Q  is  ^  1,  that  of  i?  is  —  1/3. 

B  1 


2  PLANE  ANALYTIC  GEOMETRY  [I,  §2 

2.  Determination  of  a  Point  by  its  Abscissa.  Let  us  select, 
on  a  given  line,  an  arbitrary  origin  0,  a  unit  of  measure,  and  a 
definite  sense  as  positive.  Then  any  real  number,  such  as  5, 
—  3,  7.35,  —  V2,  regarded  as  the  abscissa  of  a  point  P,  fully 
determines  the  position  of  P  on  the  line.  Conversely,  every 
point  on  the  line  has  one  and  only  one  abscissa. 

The  abscissa  of  a  point  is  usually  denoted  by  the  letter  a?, 
which,  in  analytic  geometry  as  in  algebra,  may  represent  any 
real  or  complex  number. 

To  represent  a  real  point  the  abscissa  must  be  a  real  number. 
If  in  any  problem  the  abscissa  »  of  a  point  is  not  a  real  num- 
ber, there  exists  no  real  point  satisfying  the  conditions  of  the 
problem. 

EXERCISES 

1.  What  is  the  abscissa  of  the  origin  ? 

2.  With  the  inch  as  unit  of  length,  mark  on  a  line  the  points  whose 
abscissas  are  :  3,  —2,  VS,  —  1.26,  —  \/5,  |,  —  ^. 

3.  On  a  railroad  line  running  east  and  west,  if  the  station  B  is  20  miles 
east  of  the  station  A  and  the  station  C  is  33  miles  east  of  A,  what  are  the 
abscissas  of  A  and  C  for  B  as  origin,  the  sense  eastward  being  taken  as 
positive  ? 

4.  On  a  Fahrenheit  thermometer,  what  is  the  positive  sense  ?  What 
is  the  unit  of  measure?  What  is  the  meaning  of  the  reading  66°? 
What  is  meant  by  —  7°  ? 

6.  A  water  gauge  is  a  vertical  post  carrying  a  scale  ;  the  mean  water 
level  is  generally  taken  as  origin.  If  the  water  stands  at  +  7  on  one  day 
and  at  —11  the  next  day,  the  unit  being  the  inch,  how  much  has  the 
water  fallen  ? 

6.  If  051,  X2  (read :  x  one,  x  two)  are  the  abscissas  of  any  two  points 
Pi,  P2  on  a  given  line,  show  that  the  abscissa  of  the  midpoint  between 
Pi  and  P2  is  ^  (xi  +  X2).  Consider  separately  the  cases  when  Pi,  P^  lie 
on  the  same  side  of  the  origin  0  and  when  they  lie  on  opposite  sides. 


I,  §  3]  COORDINATES  3 

3.   Ratio  of  Division.     A  segment  AB  (Fig.  2)  of  a  straight 
line  being  given,  it  is  shown  in  elementary  geometry  how  to 
find  the  point  C  that  divides   ^ 
AB  in  a  given  ratio  7c.     Thus, 
if  A;  =  I,  the  point  G  such  that 

AG^2 
AB     5 

is   found  as  follows.     On  any 

line  through  A  lay  off  AD  =  2  and  AE=5',  join  B  and  E. 
Then  the  parallel  to  BE  through  D  meets  AB  at  the  required 
point  C. 

Analytically,  the  problem  of  dividing  a  line  in  a  given  ratio 
is  solved  as  follows.  On  the  line  AB  (Fig.  3)  we  choose  a 
point  0  as  origin  and  assign  a  positive  sense.  Then  the 
abscissas  Xi  of  A  and  X2  of  B  are  known.     To  find  a  point  0 


FiG.  3 

which  divides  AB  in  the  ratio  of  division  k  =  AC/AB,  let  us 
denote  the  unknown  abscissa  of  C  by  x.     Then  we  have 

AC^x  —  x-i^,   AB  =  X2  —  x^; 
hence  the  abscissa  a;  of  O  must  satisfy  the  condition 

^  —  Xi  __^ 

•*'2  —  "^l 

whence 

X  =  Xi-\-  k  {X2  —  Xi)  ', 

or,  if  we  write  Ax  (read :  delta  x)  for  the  "  difference  of  the 
ic's,"  i.e.  Ax  z=x.2  —  Xi, 

x  =  Xi-\-k  '  Ax. 

Thus,  if  the  abscissas  of  A  and  B  are  2  and  7,  the  abscissas 


4         PLANE  ANALYTIC  GEOMETRY      [I,  §  3 

of  the  points  that  divide  AB  in  the  ratios  J,  ^,  f ,  f  are  3,  4^, 
8,  9^,  respectively.  Check  these  results  by  geometric  con- 
struction. 

If  the  segments  AC  and  AB  have  the  same  sense,  the  divi- 
sion ratio  k  is  positive.  For  example,  in  Fig.  3,  the  point  O 
lies  between  A  and  B ;  hence  the  division  ratio  A;  is  a  positive 
proper  fraction.  If  the  division  ratio  k  is  negative,  the  seg- 
ments AC  and  AB  must  have  opposite  sense,  so  that  B  and  C 
lie  on  the  opposite  sides  of  A. 

If  the  abscissas  of  A  and  B  are  again  2  and  7,  the  abscissa 
xoi  C  when  A;  =  2,  -  1,  - 1,  -  .2  will  be  12,  -  3,  0,  1,  respec- 
tively. Illustrate  this  by  a  figure,  and  check  by  the  geometric 
construction. 

4.  Location  of  a  Point  in  a  Plane.  To  locate  a  point  in 
a  plane,  that  is,  to  determine  its  position  in  a  plane,  we  may 
proceed  as  follows.  Draw  two  lines  at  right  angles  in  the 
plane ;  on  each  of  these  take  the  point  of  intersection  O  as 
origin,  and  assign  a  definite  positive  sense  to  each  line,  e.g.  by 
marking  each  line  with  an  arrowhead.  It  is  usual  to  mark 
the  positive  sense  of  one  line  by  affixing  the  letter  x  to  it,  and 
the  positive   sense   of  the   other   line  by 

affixing  the  letter  y  to   it,   as  in  Fig.  4.      ^ 

These  two  lines  are  then  called  the  axes 

of  coordinates^  or  simply  the   axes.     We    —o ~ q      ~ 

distinguish  them  by  calling  the  line  Ox  the  yiq.  4 

ic-axis,  or  axis  of  abscissas,  and  the  line  Oy 
the  ?/-axis,  or  axis  of  ordinates.  Now  project  the  point  P  on 
each  axis,  i.e.  let  fall  the  perpendiculars  PQ^  PR  from  P  on 
the  axes.  The  point  Q  has  the  abscissa  OQ  =  x  on  the  axis  Ox. 
The  point  R  has  the  abscissa  OR  =  y  on  the  axis  Oy.  The 
distance    OQ  =  RP=x   is   called   the    abscissa    of    P,   and 


I 


I,  §  6]  COORDINATES  5 

OR  =  QP  =  y  is  called  the  ordinate  of  P.  The  position  of  the 
point  P  in  the  plane  is  fully  determined  if  its  abscissa  x  and 
its  ordinate  y  are  both  given.  The  two  numbers  x,  y  are  also 
called  the  coordinates  of  the  point  P. 

5.  Signs  of  the  Coordinates.  Quadrants.  It  is  clear 
from  Fig.  4  that  x  and  y  are  the  perpendicular  distances  of  the 
point  P  from  the  two  axes.  It  should  be  observed  that  each 
of  these   numbers  may  be  positive  or 

negative,  as  in  §  1.  ,      ^ 

n  ^r" — 
The  two  axes  divide  the  plane  into  i 

four  compartments  distinguished  as  in  i 

trigonometry  as  the  first,  second,  third,  i          ^ 

and  fourth   quadrants  (Fig.  5).     It   is  ^zr  |                 jn 

readily  seen  that  any  point  in  the  first      p"' 

quadrant  has  both  its  coordinates  posi- 
tive.    What  are  the  sisrns  of  the  coordi- 


1 


■^t)' 


nates  in  the  other  quadrants  ?  What  are  the  coordinates  of  the 
origin  0  ?  What  are  the  coordinates  of  a  point  on  one  of  the 
axes  ?  It  is  customary  to  name  the  abscissa  first  and  then 
the  ordinate;  thus  the  point  (—3,  5)  means  the  point  whose 
abscissa  is  —  3  and  whose  ordinate  is  5. 

Every  point  in  the  vlanp.  hna  turn  dpfnifp  rpnl  nura^^fn  ao-rrh. 
ordinates;  conversely,  to  every  pair  of  real  numbers  correspoiids 
one  and  only  one  point  of  the  plane. 

Locate  the  points:  (6,  -2),  (0,  7),  (2-V3,  |),  (-4,  2V2), 
(-5,0). 

6.  Units.  It  may  sometimes  be  convenient  to  choose  the 
unit  of  measure  for  the  abscissa  of  a  point  different  from  the 
unit  of  measure  for  the  ordinate.  Thus,  if  the  same  unit,  say 
one  inch,  were  taken  for  abscissa  and  ordinate,  the  point  (3,  48) 
might  fall  beyond  the  limits  of  the  paper.     To  avoid  this  we 


6 


PLANE  ANALYTIC  GEOMETRY 


II,  §6 


may  lay  off  the  ordinate  on  a  scale  of  i  inch.  When  different 
units  are  used,  the  unit  used  on  each  axis  should  always  be 
indicated  in  the  drawing.  When  nothing  is  said  to  the  con- 
trary, the  units  for  abscissas  and  ordinates  are  always  under- 
stood to  be  the  same. 

7.  Oblique  Axes.  The  position  of  a  point  in  a  plane  can 
also  be  determined  with  reference  to  two  axes  that  are  not  at 
right  angles ;  but  the  angle  w  between  these 
axes  must  be  given  (Fig.  6).  The  abscissa 
and  the  ordinate  of  the  point  P  are  then 
the  segments  OQ  =  x,  OR  =  y  cut  off  on 
the  axes  by  the  parallels  through  P  to  the 
axes.     If  0)  =  ^  TT,  i.e.  if  the  axes   are   at 

right  angles,  we  have  the  case  of  rectangular  coordinates 
discussed  in  §§4,  5.  In  what  follows,  the  axes  are  always 
taken  at  right  angles  unless  the  contrary  is  definitely 
stated. 

8.  Distance  of  a  Point  from  the  Origin. 
For  the  distance  r=OP  (Fig.  7)  of  the  point 
P  from  the  origin  0  we  have  from  the  right- 
angled  triangle  OQP: 

Fig.  7 


Fig.  6 


^ 


r=  y/x^  -f  y^y 

where  x,  y  are  the  coordinates  of  P. 

If  the  axes  are  oblique  (Fig.  8),  with  the  angle 
xOy  =  CO,  we  have,  from  the  triangle  OQP,  in 
which  the  angle  at  Q  is  equal  to  ir  —  «,*  by  the 
cosine  law  of  trigonometry, 


Fig.  8 


r  —  y/x^  +  y*  —  2  xy  cos  (ir  —  «)  =  Vx-  +  y^*  -f  2  xy  cos  w. 


*  In  advanced  mathematics,  angles  are  generally  measured  in  radians,  the 
symbol  ir  denoting  an  angle  of  180^ 


I,  §  9]  COORDINATES  7 

Notice  that  these  formulas  hold  not  only  when  the  point  P 
lies  in  the  first  quadrant,  but  quite  generally  wherever  the 
point  P  may  be  situated.     Draw  the  figures  for  several  cases. 

9.  Distance  between  Two  Points.  By  Fig.  9,  the  distance 
d  =  PiP2  between  two  points  Pi{xi,  y^  and  P2(a72>  2/2)  can  be 
found  if  the  coordinates  of  the  two  points 
are  given.  For  in  the  triangle  P1QP2  we 
have 


Pi  Q  =  3^2  -  a^i ,    QP2  =  2/2  -  yi ; 
hence  F^^  9 

(1)  d  =  v/(a^2-^i)2  + (2/2-2/1)2. 

If  we  write  Ax  (§  3)  for  the  "  difference  of  the  ic's  "  and  Ay 
for  the  "  difference  of  the  y's  ",  i.e. 

Aa;  =  a;2  — a^i     and     A?/ =  2/2  — 2/1? 

the  formula  for  the  distance  has  the  simple  form 


(2)  d  =  V(Aa?)2  +  (A2/)2; 

or,  in  words, 

The  distance  between  any  two  points  is  equal  to  the  square  root 
of  the  sum  of  the  squares  of  the  differences  between  their  corre- 
sponding coordinates. 

Draw  the  figure  showing  the  distance  between  two  points 
(like  Fig.  9)  for  various  positions  of  these  points  and  show 
that  the  expression  for  d  holds  in  all  cases. 

Show  that  the  distance  between  two  points  Pi  (aji,  yi),  P2  (X2^  ^2)  when 
the  axes  are  oblique,  with  angle  w,  is 

d  =  \/{X2  -  Xl)2+  (^2  -  yiY^  +  2(X2  -  Xi){y2  -  yi)  cos  (O 

=  V(  Aa;)2  +  (A2/)2  +  2  Ax-  Ay  •  cos  w. 


8  PLANE  ANALYTIC  GEOMETRY  [I,  §  10 

10.  Ratio  of  Division.  If  two  points  P^  {x^ ,  y^  and  P^  (xo,  y^ 
are  given  by  their  coordinates^  the  coordinates  x,  y  of  any  point 
Pon  the  line  P1P2  can  be  found  if  the  division  ratio  P^P/P^P^,  =  k 
is  known  in  which  the  point  P  divides  the  segment  P1P2'  Let  Qi , 
Q2,  Q  (Fig.  10),  be  the  projections  of  P^,  P2,  P  on  the  axis  Ox-, 
then  the  point  Q  divides  Q1Q2  in  the  same  ratio  k  in  which 
P  divides  P1P2'  Now  as  OQi  =  Xi, 
0^2=  ^2?  OQ  =  a;,  it  follows  from  §  3 
that 

x=Xi  -\-k(x2  —  Xi). 


y 

R 

-        ^ 

-A 

R, 

--X 

1       1     X 

0 

X         «r 

Q  q. 

In  the  same  way  we  find  by  projecting 
Pi,  P2,  P on.  the  axis  Oy  that 

FiQ.  10 

y  =  yi  +  ^(y2-yi)' 

Thus,  the  coordinates  x,  y  oi  P  are  found  expressed  in  terms 
of  the  coordinates  of  Pj ,  Po  and  the  division  ratio  k.  Putting 
again  X2  —  Xi  =  Ax,y2  —  yi=Ay,  we  may  also  write 

x  =  Xi-\-k'  /iix,     y  =  yi-\-k'Ay. 

Here  again  the  student  should  convince  himself  that  the 
formulas  hold  generally  for  any  position  of  the  two  points,  by 
selecting  numerous  examples.  He  should  also  prove,  from  a 
figure,  that  the  same  expressions  for  the  coordinates  of  the 
point  P  hold  for  oblique  coordinates. 

As  in  §  3,  if  the  division  ratio  k  is  negative,  the  two 
segments  P1P2  and  PiP  must  have  opposite  sense,  so  that 
the  points  P  and  P2  must  lie  on  opposite  sides  of  the 
point  Pj. 

Find,  e.g.,  the  coordinates  of  the  points  that  divide  the  seg- 
ment joining  (—  4,  3)  to  (6,  —  5)  in  the  division  ratios  k  =  ^, 
k=2,  A:=— i,  k  =  —l,  and  indicate  the  four  points  in  a 
fiGTure. 


I,  §  11]  COORDINATES  9 

11.  Midpoint  of  a  Segment.  The  midpoint  P  of  a  segment 
P1P2  has  for  its  coordinates  the  arithmetic  means  of  the  corre- 
sponding coordinates  of  Pi  and  P^ ;  that  is,  if  x^ ,  y^  are  the  co- 
ordinates of  Pi,  ^2,  2/2  those  of  Pg)  the  division  ratio  being 
A;  =  i   the  coordinates  of  the  midpoint  P  are  (§  10) 

a;  =  a?i  +  1(0^2  -  ajj)  =  Ka^i  +  ^2), 

2/  =  2/i  +  i(2/2  -  2/1)  =  }(2/i  +  2/2). 

EXERCISES 

1.  With  reference  to  the  same  set  of  axes,  locate  the  points  (6,  4), 
(2,  -  i),  (-  6.4,  -  3.2),  (-4,  0),  (-  1,  5),  (.001,  -  4.01). 

2.  Locate  the  points  (-3,4),  (0,-1),  (6,  -  V2),  (f^  _  10|), 
(0,a),  (a,  6),  (3,  -2),  (-2,  V2). 

3.  If  a  and  h  are  positive  numbers,  in  what  quadrants  do  the  follow- 
ing points  lie  :  (a,  —  &),  (6,  a),  (a,  a),  (-  6,  6),  (—6,  —  a)? 

4.  Show  that  the  points  (a,  &)  and  («,  —  6)  are  symmetric  with 
respect  to  the  axis  Ox ;  that  (a,  &)  and  (—a,  6)  are  symmetric  with  re- 
spect to  the  axis  Oy  \  that  (a,  &)  and  (—  a,  —  6)  are  symmetric  with 
respect  to  the  origin. 

6.  In  the  city  of  Washington  the  lettered  streets  (A  street,  B  street, 
etc.)  run  east  and  west,  the  numbered  streets  (1st  street,  2d  street,  etc.) 
north  and  south,  the  Capitol  being  the  origin  of  coordinates.  The  axes 
of  coordinates  are  called  avenues,  thus,  e.gr.,  1st  street  runs  one  block 
east  of  the  Capitol.  If  the  length  of  a  block  were  1/10  mile,  what  would 
be  the  distance  from  the  corner  of  South  C  street  and  East  5th  street  to 
the  corner  of  North  Q  street  and  West  14th  street  ? 

6.  Prove  that  the  points  (6,  2),  (0,  —  6),  (7,  1)  lie  on  a  circle  whose 
center  is  (3,  —  2). 

7.  A  square  of  side  s  has  its  center  at  the  origin  and  diagonals  coin- 
cident with  the  axes  ;  what  are  the  coordinates  of  the  vertices  ?  of  the 
midpoints  of  the  sides  ? 

8.  If  a  point  moves  parallel  to  the  axis  0^,  which  of  its  coordinates 
remains  constant  ? 


10  PLANE  ANALYTIC  GEOMETRY  [I,  §  11 

9.  In  what  quadrants  can  a  point  lie  if  its  abscissa  is  negative  ?  its 
ordinate  positive  ? 

10.  Find  the  coordinates  of  the  points  which  trisect  the  distance  be- 
tween the  points  (1,  —  2)  and  (—  3,  4). 

11.  To  what  point  must  the  line  segment  drawn  from  (2,  —3)  to 
(—3,  6)  be  extended  so  that  its  length  is  doubled  ?  trebled  ? 

12.  The  abscissa  of  a  iwint  is  —  3,  its  distance  from  the  origin  is  5 ; 
what  is  its  ordinate  ? 

13.  A  rectangular  house  is  to  be  built  on  a  comer  lot,  the  front,  30  ft. 
wide,  cutting  off  equal  segments  on  the  adjoining  streets.  If  the  house  is 
20  ft.  deep,  find  the  coordinates  (with  respect  to  the  adjoining  streets)  of 
the  back  comers  of  the  house. 

14.  A  baseball  diamond  is  90  ft.  square  and  pitcher's  plate  is  60  ft. 
from  home  plate.  Using  the  foul  lines  as  axes,  find  the  coordinates  of 
the  following  positions  : 

(a)  pitcher's  plate  ; 

(6)  catcher  8  ft.  back  of  home  plate  and  in  line  with  second  base  ; 

(c)  base  runner  playing  12  ft.  from  first  base  ; 

(d)  third  baseman  playing  midway  between  pitcher's  plate  and  third 
base  (before  a  bunt) ; 

(c)  right  fielder  playing  90  ft.  from  first  and  second  base  each. 

16.  How  far  does  the  ball  go  in  Ex.  14  if  thrown  by  third  baseman 
in  position  (d)  to  second  base  ? 

16.  If  right  fielder  (Ex.  14)  catches  a  ball  in  position  (e)  and  throws 
it  to  third  base  for  a  double  play,  how  far  does  the  ball  go  ? 

17.  A  park  600  ft.  long  and  400  ft.  wide  has  six  lights  arranged  in  a 
circle  about  a  central  light  cluster.  All  the  lights  are  200  ft.  apart,  and 
the  central  cluster  and  two  others  are  in  a  line  parallel  to  the  length  of 
the  park.  What  are  the  coordinates  of  all  the  Ughts  with  respect  to  two 
boundary  hedges  ? 

18.  With  respect  to  adjoining  walks,  three  trees  have  coordinates 
(30  ft.,  8  ft.),  (20  ft.,  45  ft.),  (-  27  ft.,  14  ft,),  respectively.  A  tree  is  to 
be  planted  to  form  the  fourth  vertex  of  a  parallelogram;  where  should  it 
be  placed  ?     (Three  possible  positions ;  best  found  by  division  ratio.) 


I,  §  13] 


COORDINATES 


11 


12.   Area  of  a  Triangle  with  One  Vertex  at  the  Origin. 

Let  one  vertex  of  a  triangle  be  the  origin,  and  let  the  other 
vertices  be  Pi(a;„  2/1)  and  P2(a^2>  2/2)-  Draw  through  P^  and 
P2  lipes  parallel  to  the  axes  (I'ig.  11).  The 
area  A  of  the  triangle  is  then  obtained  by- 
subtracting  from  the  area  of  the  circum- 
scribed rectangle  the  areas  of  the  three  non- 
shaded  triangles ;  i.e. 

A^x^y^  -  l^i?/!  -  |-i»22/2  -  K^i  -«^2)(2/2  -2/1) 

=  i(^iy2  -  a^22/i) ; 

or,  in  determinant  notation, 

X,     Pi 


Fig.  11 


X2     2/2 


This  formula  gives  the  area  with  the  sign  -f  or  —  according 
as  the  sense  of  the  motion  around  the  perimeter  OP^P^O  is 
counterclockwise  (opposite  to  the  rotation  of  the  hands  of  a 
clock)  or  clockwise. 

13.  Translation  of  Axes.  Instead  of  the  origin  0  and  the 
axes  Ox,  Oy  (Fig.  12),  let  us  select  a  new  origin  0'  (read :  0 
prime)  and  new  ax:es  O'x',  O'y',  parallel  to  the  old  axes.  Then 
any  point  P  whose  coordinates  with  reference  to  the  old  axes 
are  OQ  =  Xj  QP  =  y  will  have  with 
reference  to  the  new  axes  the  coordi- 
nates 0'Q'  =  x',  Q'P=y';  and  the 
figure  shows  that  if  h,  k  are  the  co- 
ordinates of  the  new  origin,  then 

x  =  x'  -]-h, 
y  =  y'  +  k. 


0^ 


y1 


Fig.  12 


The  change  from  one  set  of  axes  to  a  new  set  is  called  a 
transformation  of  coordinates.     In  the  present  case,  where  the 


12 


PLANE  ANALYTIC  GEOMETRY 


[I,  §  13 


new  axes  are  parallel  to  the  old,  this  transformation  can  be 
said  to  consist  in  a  translation  of  the  axes. 

14.  Area  of  Any  Triangle.  Let  Py{x^,  y^),  P2(^2)2/2), 
P^ix^,  2/3)  he  the  vertices  of  the  triangle  (Fig.  13).  If  we  take 
one  of  these  vertices,  say  Pg,  as  new 
origin,  with  the  new  axes  parallel  to  the 
old,  the  new  coordinates  of  Pi ,  Pg  will  be : 

3/ 2 Xi  fl/g,      X  2  — ~  X2 fl/g, 

y\=yi-y3,  ^2= 2/2 -2/3. 

Hence,  by  §  12,  the  area  of  the  triangle 

=  K^i{y2  -  2/3)  +  3:2(2/3  -  2/1) + 2:3(2/1  -  2/2)] ; 

or,  in  determinant  notation, 


^=i 


^l 

yi 

1 

X2 

2/2 

1 

X3 

2/3 

1 

Here  as  in  §  12  the  sign  of  the  area  is  +  or  —  according  as 
the  sense  of  the  motion  along  the  perimeter  P1P2P3P1  is  coun- 
terclockwise. 

EXERCISES 

1.  Find  the  areas  of  the  triangles  having  the  following  vertices  : 

(a)  (1,  3),  (5,  2),  (4,  6)  ;  (6)  (-  2,  1),  (2,  -  3),  (0,  -  6)  ; 

(c)    (a,  &),  (a,  0),  (0,  6)  ;  (d)  (4,  3),  (6,  -  2),  (-  1,  6). 

2.  Show  that  the  area  of  the  triangle  whose  vertices  are  (7,  —8), 
(—3,  2),  (—  5,  —4)  is  four  times  the  area  of  the  triangle  formed  by- 
joining  the  midpoints  of  the  sides. 

3.  Find  the  area  of  the  quadrilateral  whose  vertices  are  (2,  3),  (—  1, 
-1),  (-4,2),  (-3,6). 


I,  §  15] 


COORDINATES 


13 


4.  Find  the  area  of  the  triangle  whose  vertices  are  (a,  0),  (0,  &), 
(-C,  -c). 

5.  Find  the  area  of~^he  triangle  (1,  4),  (3,  -2),  (-3,  16).  What 
does  your  result  show  about  these  points  ? 

6.  Find  the  area  of  the  triangle  (a,  6  +  c),  (b^  c-\-  a),  (c,  a  +  &)• 
What  does  the  result  show  whatever  the  values  of  a,  &,  c  ? 

7.  Show  that  the  points  (3,  7),  (7,  3),  (8,  8)  are  the  vertices  of  an 
isosceles  triangle.  What  is  its  area  ?  Show  that  the  same  is  true  for  the 
points  (a,  6),  (&,  a),  (c,  c),  whatever  a,  6,  c,  and  find  the  area. 

8.  Find  the  perimeter  of  the  triangle  whose  vertices  are  (3,  7),  (2, 
—  1),  (5,  3).     Is  the  triangle  scalene  ?     What  is  its  area  ? 

9.  Show  that  the  area  of  a  quadrilateral  whose  vertices  are  (oji,  yi), 
(a^2,  2/2),  (xs,  2/3),  (Xi,  y^)  may  be  written  in  the  form 


A  =  l 


x\  —  xz    yi  —  2/3 
X2  —  X4    yi  —  yi 


15.  Statistics.  Related  Quantities.  If  pairs  of  values 
of  two  related  quantities  are  given,  each  of  these  pairs  of 
values  is  represented  by  a  point  in  the  plane  if  the  value  of 
one  quantity  is  represented  by  the  abscissa  and  that  of  the 
other  by  the  ordinate  of  the  point.  A  curved  line  joining 
these  points  gives  a  vivid  idea  of  the  way  in  which  the  two 
quantities  change.  Statistics  and  the  results  of  scientific  ex- 
periments are  often  represented  in  this  manner. 

EXERCISES 
1.  The  population  of  the  United  States,  as  shown  by  the  census  reports, 
is  approximately  as  given  in  the  following  table  : 


Teak 

1800 

'10 

'20 

'30 

'40 

'50 

'60 

'70 

'80 

'90 

1900 

'10 

Million 

4 

5 

7 

10 

13 

17 

23 

31 

■ 

39 

50 

63 

76 

92 

14 


PLANE  ANALYTIC  GEOMETRY 


[I,  §  15 


Mark  the  points  corresponding  to  the  pairs  of  numbers  (1790,  4), 
(1800,  5),  etc.,  on  squared  paper,  representing  the  time  on  the  horizontal 
axis  and  the  population  vertically.     Connect  these  points  by  a  curved  line. 

2.  From  the  figure  of  Ex.  1,  estimate  approximately  the  population  of 
the  United  States  in  1875  ;  in  1905  ;  in  1915. 

3.  From  the  figure  of  Ex.  1,  estimate  approximately  when  the  popula- 
tion was  25  millions  ;  60  millions  ;  when  it  will  be  100  millions. 

4.  Draw  a  figure  to  represent  the  growth  of  the  population  of  your 
own  State,  from  the  figures  given  by  the  Census  Reports. 

[Other  data  suitable  for  statistical  graphs  can  be  found  in  large  quan- 
tity in  the  Census  Reports ;  in  the  Crop  Reports  of  the  government ;  in 
the  quotations  of  the  market  prices  of  food  and  of  stocks  and  bonds  ;  in 
the  World  Almanac  ;  and  in  many  other  books.  ] 

6.   The  temperatures  on  a  certain  day  varied  hour  by  hour  as  follows : 


A.M. 

N. 

P.M. 

Time  .  . 
Temp.  .  . 

6 
50 

7 
62 

8 
55 

9 
60 

10 
64 

11 

67 

12 
70 

1 

72 

2 
74 

3 

75 

4 

74 

5 
72 

6 
69 

7 
65 

8 
60 

9 
57 

Draw  a  figure  to  represent  these  pairs  of  values. 

6.  In  experiments  on  stretching  an  iron  bar,  the  tension  t  (in  tons) 
and  the  elongation  E  (in  thousandths  of  an  inch)  were  found  to  be  as 
follows  : 


t  (in  tons) 

E  (in  thousandths  of  an  inch) 


1 
10 


2 
19 


4 

38 


6 
60 


10 
103 


Draw  a  figure  to  represent  these  pairs  of  values. 
[Other  data  can  be  found  in  books  on  Physics  and  Engineering.] 
7.  By  Hooke's  law,  the  elongation  ^  of  a  stretched  rod  is  supposed 
to  be  connected  with  the  tension  t  by  the  formula  E  =  c  -  t^  where  c  is  a 
constant.  Show  that  if  c  =  10,  with  the  units  of  Ex.  6,  the  values  of  E 
and  t  would  be  nearly  the  same  as  those  of  Ex.  6.  Plot  the  values  given 
by  the  formula  and  compare  with  the  figure  of  Ex.  6. 


I,  §  16] 


COORDINATES 


15 


8.  The  distances  through  which  a  body  will  fall  from  rest  in  a  vacuum 
in  a  time  t  are  given  by  the  formula  s  =  16 1'^,  approximately,  if  t  is  in 
seconds  and  s  is  in  feet.     Show  that  corresponding  values  of  s  and  t  are 


2 

64 


3 
144 


4 

256 


6 
400 


6 

576 


Draw  a  figure  to  represent  these  pairs  of  values. 

16.  Polar  Coordinates.  The  position  of  a  point  P  in  a 
plane  (Fig.  14)  can  also  be  assigned  by  its  distance  OP  =  r 
from  a  fixed  point,  or  pole,  0,  and  the  angle  xOP  =  <^,  made 
by  the  line  OP  with  a  fixed  line  Ox,  the  polar  axis.  The  dis- 
tance r  is  called  the  radius  vector,  the  angle  <^  the  polar  angle 
(or  also  the  vectorial  angle,  azimuth,  ampli-  j, 

tude,  or  anomaly)  of   the   point  P.     The  /r^'' 

radius  vector  r  and  the  polar  angle  eft  are     q^^^^ je^ 

called  the  polar  coordinates  of  P.  Fig.  14 

Locate  the  points :  (5,  |  tt),  (6,  |  tt),  (2,  140°),  (7,  307°), 
(V5,  tt),  (4,  0°). 

To  obtain  for  every  point  in  the  plane  a  single  definite  pair  of  polar 
coordinates  it  is  sufficient  to  take  the  radius  vector  r  always  positive  and 
to  regard  as  polar  angle  the  positive  angle  between  0  and  27r(O^0<27r) 
through  which  the  polar  axis  (regarded  as  a  half-line  or  ray  issuing  from 
the  pole  0)  must  be  turned  about  the  pole  0  in  the  counterclockwise 
sense  to  pass  through  P.  The  only  exception  is  the  pole  0  for  which 
r  =  0,  while  the  polar  angle  is  indeterminate. 

But  it  is  not  necessary  to  confine  the  radius  vector  to  positive  values 
and  the  polar  angle  to  values  between  0  and  2  tt.  A  single  definite  point 
P  will  correspond  to  every  pair  of  real  values  of  r  and  0,  if  we  agree  that 
a  negative  value  of  the  radius  vector  means  that  the  distance  r  is  to  be 
laid  off  in  the  negative  sense  on  the  polar  axis,  after  being  turned  through 
the  angle  0,  and  that  a  negative  value  of  <p  means  that  the  polar  axis 
should  be  turned  in  the  clockwise  sense. 

The  polar  angle  is  then  not  changed  by  adding  to  it  any  positive  or 


16  PLANE  ANALYTIC  GEOMETRY  [I,  §  16 

negative  integral  multiple  of  2  tt  ;  and  a  point  whose  polar  coordinates  are 
r,  0  can  also  be  described  as  having  the  coordinates  —  r,  ^  ±  t. 

Locate  the  points : 

(3,  -i^),  {a,  -  |,r),  (-  5,  75°),  (-  3,  -  20°). 

17.  Transformation  from  Cartesian  to  Polar  Coordinates, 

and  vice  versa.  The  coordinates  OQ  =  x,  QP  =  y,  defined  in 
§  4,  are  called  cartesian  coordinates,  to  distinguish  them 
from  the  polar  coordinates.  The  term  is  derived  from  the 
Latin  form,  Cartesius,  of  the  name  of  Rene  Descartes,  who 
first  applied  the  method  of  coordinates  systematically  (1637), 
and  thus  became  the  founder  of  analytic  geometry. 

The  relation  between  the  cartesian  and  polar  coordinates  of 
one   and   the   same   point   P  appears   from      V 
Fig.  15.     We  have  evidently : 

x  =  r  cos  <f>,         , 
.  and 

y  =  r8in<f>, 

18.  Distance  between  Two  Points  in  Polar  Coordinates. 

If  two  points  Pi,  P2  are  given  by  their  polar  coordinates,  r,, 
^  and   ra,  <^2>  the  distance  d  =  P^P^  between 
them  is  found  from  the  triangle  OP^P^  (Fig.  16), 
by  the  cosine  law  of  trigonometry,  if  we  ob- 
serve that  the  angle  at  0  is  equal  to  ±  {<t>2—<t>i) '- 


y- 

p 

tan  <f}  =z^» 

X 

^^          X             1        X 

d 

Q 
Fia.  15 

d  =  Vri"  +  ra^  -  2  r,r^  cos  {<t>o  -  <^i).  ^     fiq.  16 


EXERCISES 

1.  Find  the  distances  between  the  points:   (2,  lir)   and   (4,  fTr); 
(a,  iir)  and  (3  a,  ^tt). 

2.  Find  the  cartesian  coordinates  of  the  points  (5,  ^ir),  ("6,  —  i  tt), 
(4,  i  n),  (2,  I  ,r),  (7,  tt),  (6,  -  x),  (4,  0),  (-  3,  60°),  (-  6,  -  90°)  . 


I,  §  19] 


COORDINATES 


17 


3.  Find  the  polar  coordinates  of  the  points  (VS,  1),  (-VS,  1), 
(1,-1),  (-1,-1),  (-a,  a). 

4.  Find  an  expression  for  the  area  of  a  triangle  whose  vertices  are 
(0,  0),  (n,  00,  and  (ra,  02). 

5.  Find  the  area  of  the  triangle  whose  vertices  are  (n,  0i),  {1%  02), 

(»*3,  03). 

19.  Projection  of  Vectors.  A  straight  line  segment  AB 
of  definite  length,  direction,  and  sense  (indicated  by  an  arrow- 
head, pointing  from  A  to  B)  is  called  a  vector.  The  projection 
A'B'  (Figs.  17,  18)  of  a  vector  AB  on  an  axis,  i.e.  on  a  line  I 

B 


1 

1 

'^___ 

^  1 
1 
.__±_ 

1 

1 

-> 

A' 

I 

B' 

Fig. 

17 

on  which  a  definite  sense  has  been  selected  as  positive,  is  the 
product  of  the  length  of  the  vector  AB  into  the  cosine  of  the  angle 
between  the  positive  senses  of  the  axis  and  the  vector : 

A'B'  =  AB  cos  a. 

The  positive  sense  of  the  axis  makes  with  the  vector  two  angles 
whose  sum  is  2  tt  =  360°.  As  their  cosines  are  the  same,  it 
makes  no  difference  which  of 
the  two  angles  is  used. 

With  these  conventions  it  is 
readily  seen  that  the  sum  of  the 
projections  of  the  sides  of  an 
open  polygon  on  any  axis  is  equal 
to  the  projection  of  the  closing  side 
on  the  same  axis,  the  sides  of  the 
open  polygon  being  taken  in  the  same  sense  around  the  perim- 
eter. 

c 


Thus,  in  Fig.  19,  the  vectors  P^P^,  P^P^,  •••  P^Pq  are  in- 


18 


PLANE  ANALYTIC  GEOMETRY 


[I,  §  19 


clined  at  the  angles  cti,  a2,  •••  a^  to  the  axis  I ;  the  closing  line  PiP^ 

makes  the  angle  a  with  I ;  its  projection  is  P'iP'q  ;  and  we  have 

P1P2  cos  tti+PgA  cos  02+^3 A  cos  as-\-P^Ps  cos  u^-^-P^Pq  cos  a^ 

=  P'lP'e  =  PiPg  cos  a. 
For,  if  the  abscissas  of  Pi,  P2,  •••  Pe  measured  along  I,  from 
any   origin    0  on   Z,  are   Xi,  X2,  •••  Xg,  the   projections  of   the 
vectors  are  X2  —  Xi,  x^  —  X2,  etc.,  so  that  our  equation  becomes 
the  identity : 

X^-Xi  +  Xs  —  X2-\-X^-Xs+Xs-X^  +  Xe-Xs  =  Xfi-Xi. 

20.  Components  and  Resultants  of  Vectors.  In  physics, 
forces,  velocities,  accelerations,  etc.,  are  represented  by  vectors 
because  such  magnitudes  have  not  only  a  numerical  value  but 
also  a  definite  direction  and  sense. 

According  to  the  parallelogram  law  of  physics,  two  forces  OPi, 
OP2,  acting  on  the  same  particle,  are  together  equivalent  to 
the  single  force  OP  (Eig.  20),  whose  vector 
is  the  diagonal  of  the  parallelogram  formed 

with  OPi,  OP2  as  adjacent  sides.    The  same   q^.^^^ ^^ 

law  holds  for  simultaneous  velocities  and  Fiq-  20 

accelerations,  and  for  simultaneous  or  consecutive  rectilinear 
translations.  The  vector  OP  is  called  the  resultant  of  OPi  and 
OP2,  and  the  vectors  OPi,  OP2  are  called  the  components 
of  OP. 

To  construct  the  result- 
ant it  suffices  to  lay  off  from 
the  extremity  of  the  vector 
OPi  the  vector  P^P  =  OP2 ; 
the  closing  line  OP  is  the  ^ 
resultant.  This  leads  at 
once  to  finding  the  result- 
ant OP  of  any  number  of 


Fig.  21 


I,  §  20]  COORDINATES  19 

vectors,  by  adding  the  component  vectors  geometrically,  i.e. 
putting  them  together  endwise  successively,  as  in  Fig.  21, 
where  the  dotted  lines  need  not  be  drawn. 

By  §  19,  the  projection  of  the  resultant  on  any  axis  is  equal 
to  the  sum  of  the  projections  of  all  the  components  on  the 
same  axis. 

EXERCISES 

1.  The  cartesian  coordinates  x,  y  of  any  point  P  are  the  projections  of 
its  radius  vector  OP  on  the  axes  Ox,  Oy.     (See  §  16.) 

2.  The  projection  of  any  vector  AB  on  the  axis  Ox  is  the  difference 
of  the  abscissas  of  A  and  B  ;  similarly  for  Oy. 

3.  A  force  of  10  lb.  is  inclined  to  the  horizon  at  60° ;  find  its  hori- 
zontal and  vertical  components. 

4.  A  ship  sails  40  miles  N.  60°  E.  then  24  miles  N.  45°  E.  How  far 
is  the  ship  then  from  its  starting  point  ?    How  far  east  ?    How  far  north  ? 

6.  A  point  moves  5  ft.  along  one  side  of  an  equilateral  triangle,  then 
6  ft.  parallel  to  the  second,  and  finally  8  ft.  parallel  to  the  third  side. 
What  is  the  distance  from  the  starting  point  ? 

6.  The  sum  of  the  projections  of  the  sides  of  any  closed  polygon  on  any 
axis  is  zero. 

7.  If  three  forces  acting  on  a  particle  are  parallel  and  proportional  to 
the  sides  of  a  triangle,  the  forces  are  in  equilibrium,  i.e.  their  resultant  is 
zero.     Similarly  for  any  closed  polygon. 

8.  Find  the  resultant  of  the  forces  OPi,  OP2,  OP3,  OP4,  OP5,  if 
the  coordinates  of  Pi,  P2,  P3,  P4,  P5,  with  0  as  origin,  are  (3,  1), 
(1,  2),  (-1,  3),  (-2,  -2),  (2,  -2).  (Resolve  each  force  into  its 
components  along  the  axes.) 

9.  If  any  number  of  vectors  (in  the  same  plane),  applied  at  the  ori- 
gin, are  given  by  the  coordinates  x,  y  of  their  extremities,  the  length  of 


the  resultant  is  =  \/(Sx)2  +(2^)2  (where  Sa;  means  the  sum  of  the  ab- 
scissas, Sy  the  sum  of  the  ordinates),  and  its  direction  makes  with  Ox  an 
angle  a  such  that  tan  a  =  Sy/Sx. 

10.   Find  the  horizontal  and  vertical  components  of  the  velocity  of  a 
ball  when  moving  200  ft. /sec.  at  an  angle  of  30°  to  the  horizon. 


20  PLANE  ANALYTIC  GEOMETRY  [I,  §  20 

11.  Six  forces  of  1,  2,  3,  4,  5,  6  lb.,  making  angles  of  60°  each  with 
the  next,  are  applied  at  the  same  point,  in  a  plane  ;  find  their  resultant. 

12.  A  particle  at  one  vertex  of  a  square  is  acted  upon  by  three  forces 
represented  by  the  vectors  from  the  particle  to  the  other  three  vertices ; 
find  the  resultant. 

21.  Geometric  Propositions.  In  using  analytic  geometry 
to  prove  general  geometric  propositions,  it  is  generally  conven- 
ient to  select  as  origin  a  prominent  point  in  the  geometric 
figure,  and  as  axes  of  coordinates  prominent  lines  of  the  figure. 

Example.  In  any  right  triangle,  the  distance  from  the 
vertex  of  the  right  angle  to  the  midpoint  of  the  hypotenuse 
is  equal  to  half  the  hypotenuse. 

Since  this  theorem  is  true,  if  at  all,  when  the  triangle  is  in 
any  position,  we  may  place  the  vertex  of  the  right  angle  at  the 
origin  and  the  adjacent  sides  along  the  positive  axes.  Then  the 
coordinates  of  the  vertices  are  (0,  0),  (a,  0),  (0,  &),  where  a  and 
b  are  the  lengths  of  the  two  sides  about  the  right  angle. 

The  length  of  the  hypotenuse  is  Va^  -j-  b^.  The  midpoint  of 
the  hypotenuse  has  the  coordinates  (a/2,  6/2),  by  §  11.  Hence 
the  distance  from  this  point  to  the  vertex  of  the  right  angle 
(0, 0)  is  V(a/2)2+ (6/2)2  ^  i. VoH^-  Since  this  is  half  the 
length  of  the  hypotenuse,  the  theorem  is  proved. 

Sometimes  greater  symmetry  and  elegance  is  gained  by  tak- 
ing the  coordinate  system  in  a  general  position. 

MISCELLANEOUS   EXERCISES 

1.  A  regular  hexagon  of  side  1  has  its  center  at  the  origin  and  one 
diagonal  coincident  with  the  axis  Ox ;  find  the  coordinates  of  the  vertices. 

2.  If  a  square,  with  each  side  6  units  in  length,  is  placed  with  one 
vertex  at  the  origin  and  a  diagonal  coincident  with  the  axis  Ox,  what  are 
the  coordinates  of  the  vertices  ? 

3.  If  a  rectangle,  with  two  sides  3  units  in  length  and  two  sides 
3V3  units  in  length,  is  placed  with  one  vertex  at  the  origin  and  a  diagonal 


I,  §21]  COORDINATES  21 

along  the  axis  Ox,  what  are  the  coordinates  of  the  vertices  ?    There  are 
two  possible  positions  of  the  rectangle  ;  give  the  answers  in  both  cases. 

4.  Show  that  the  points  (0,  -  1),  (-  2,  3),  (6,  7),  (8,  3)  are  the 
vertices  of  a  parallelogram.     Prove  that  this  parallelogram  is  a  rectangle. 

5.  Show  that  the  points  (1,  1),  (-  1,  -  1),  (  +  V3,  -  V3)  are  the 
vertices  of  an  equilateral  triangle. 

6.  Show  that  the  points  (6,  6),  (3/2,  -  3),  (-3,  12),  (-  \S  3)  are 
the  vertices  of  a  parallelogram. 

7.  Find  the  radius  and  the  coordinates  of  the  center  of  the  circle  pass- 
ing through  the  three  points  (2,  3),  (-  2,  7),  (0,  0). 

8.  The  vertices  of  a  triangle  are  (0,  6),  (4,  -  3),  (—  5,  6).  Find  the 
lengths  of  the  medians  and  the  coordinates  of  the  centroid  of  the  triangle, 
i.e.  of  the  intersection  of  the  medians. 

Prove  the  following  propositions  : 

9.  The  diagonals  of  any  rectangle  are  equal. 

10.  The  distance  between  the  midpoints  of  two  sides  of  any  triangle  is 
equal  to  half  the  third  side. 

11.  The  distance  between  the  midpoints  of  the  non-parallel  sides  of  a 
trapezoid  is  equal  to  half  the  sum  of  the  parallel  sides. 

12.  The  line  segments  joining  the  midpoints  of  the  adjacent  sides  of  a 
quadrilateral  form  a  parallelogram. 

13.  If  two  medians  of  a  triangle  are  equal,  the  triangle  is  isosceles. 

14.  In  any  triangle  the  sum  of  the  squares  of  any  two  sides  is  equal 
to  twice  the  square  of  the  median  drawn  to  the  midpoint  of  the  third  side 
plus  half  the  square  of  the  third  side. 

15.  The  line  segments  joining  the  midpoints  of  the  opposite  sides  of 
any  quadrilateral  bisect  each  other. 

16.  The  sum  of  the  squares  of  the  sides  of  a  quadrilateral  is  equal  to 
the  sum  of  the  squares  of  the  diagonals  plus  four  times  the  square  of  the 
line  segment  joining  the  midpoints  of  the  diagonals. 

17.  The  difference  of  the  squares  of  any  two  sides  of  a  triangle  is  equal 
to  the  difference  of  the  squares  of  their  projections  on  the  third  side. 

18.  The  vertices  (a:i,  yi),  (x^^  1/2),  (xs^  y^)  of  a  triangle  being  given, 
find  the  centroid  (intersection  of  medians). 


CHAPTER  II 


THE  STRAIGHT  LINE 


M- 


22.    Line  Parallel  to  an  Axis.    When  the  coordinates  a;,  y 
of  a  point  P  with  reference  to  given  axes  Ox,  Oy  are  known, 
the  position  of  P  in  the  plane  of  the  axes  is  determined  com- 
pletely and   uniquely.      Suppose   now 
that   only  one   of  the   coordinates    is 
given,   say,  a;  =  3 ;    what   can  be   said 
about   the   position   of   the   point   P? 
It   evidently   lies   somewhere    on   the 
line  AB  (Fig.  22)   that  is  parallel  to 
the   axis    Oy  and  has  the  distance  3 
from  Oy.     Every  point  of  the  line  AB 
has  an  abscissa  a;  =  3,  and  every  point 
whose  abscissa  is  3  lies  on  the  line  AB.     For  this  reason  we 

say  that  the  equation 

a;  =  3 

represents  the  line  AB;  we  also  say  that  a;  =  3  is  the  equation 
of  the  line  AB. 

More  generally,  the  equation  x  =  a,  where  a  is  any  real 
number,  represents  that  parallel  to  the  axis  Oy  whose  distance 
from  Oy  is  a.  Similarly,  the  equation  y  =  b  represents  a 
parallel  to  the  axis  Ox. 


Fig.  22 


EXERCISES 
Draw  the  lines  represented  by  the  equations : 

1.  x=-2.  4.  5 a:  =  7. 

2.  x  =  0.  6.   y  =  0. 

3.  x  =  12.6.  6.    2y=-7. 

22 


Sx+l  = 
10-Sy 
y=±2. 


II,  §24]  THE  STRAIGHT  LINE  23 

23.  Line  through  the  Origin.  Let  us  next  consider  any 
line  *  through  the  origin  0,  such  as  the  line  OP  in  Fig.  23. 
The  points  of  this  line  have  the  prop-  « 
erty  that  the  ratio  y/x  of  their  coordi- 
nates is  the  same,  wherever  on  this 
line  the  point  P  be  taken.  This  ratio 
is  equal  to  the  tangent  of  the  angle  a 
made  by   the  line   with  the  axis   Ox,  Fig.  23 

i.e.  to  what  we  shall  call  the  slope  of  the  line.     Let  us  put 

tan  a  =  m  ; 

then  we  have,  for  any  point  P  on  this  line  :  y/x  =  m,  i.e. : 

(1)  y  =  mx. 

Moreover,  for  any  point  Q,  not  on  this  line,  the  ratio  y/x 
must  evidently  be  different  from  tan  a,  i.e.  from  m.  The  equa- 
tion y  =  mx  is  therefore  said  to  represent  the  line  through  0 
whose  slope  is  m;  and  y  =  mx  is  called  the  equation  of  this  line. 
We  mean  by  this  statement  that  the  relation  y  —  mx  is  satis- 
fied by  the  coordinates  of  every  point  on  the  line  OP,  and  only 
by  the  coordinates  of  the  points  on  this  line.  Notice  in  partic- 
ular that  the  coordinates  of  the  origin  0,  i.e.  a;  =  0,  y  =  0, 
satisfy  the  equation  y  =  mx. 

24.  Proportional  Quantities.  Any  two  values  of  x  are 
proportional  to  the  corresponding  values  ofyiiy=  mx.  For, 
if  (a?!,  2/i)  and  {xz,  2/2)  are  two  pairs  of  values  of  x  and  y  that 
satisfy  (1),  we  have 

yi  =  mxi,  y2  =  mx2; 
hence,  dividing, 

2/1/^2  =  a;i/iC2- 

*  For  the  sake  of  brevity,  a  straight  line  will  here  in  general  be  spoken  of 
simply  as  a  line  ;  a  line  that  is  not  straight  will  be  called  a  curve. 


24  PLANE  ANALYTIC  GEOMETRY  [II,  §  24 

The  constant  quantity  m  is  called  the  factor  of  propoHionality. 

Many  instances  occur  in  mathematics  and  in  the  applied 
sciences  of  two  quantities  related  to  each  other  in  this  man- 
ner. It  is  often  said  that  one  quantity  y  varies  as  the  other 
quantity  x. 

Thus  Hooke's  Law  states  that  the  elongation  ^  of  a  stretched 
wire  or  spring  varies  as  the  tension  t ;  that  is,  E  =  kt^  where  k 
is  a  constant. 

Again,  the  circumference  c  of  a  circle  varies  as  the  radius  r ; 

that  is, 

c  =  2  Trr. 

EXERCISES 

1.  Draw  each  of  the  lines  : 

(a)y  =  2x.  (c)y=-^^x.        (e)5x4-3y  =  0.       (g)y=-x. 

(b)y=-3x.       (d)  6y  =  Sx.         {f)y  =  x.  (h)  x  -  y  =  0. 

2.  Show  that  the  equation  ax  -\-  by  =  0  can  be  reduced  to  the  form 
y  =  mx,  if  &  T^  0,  and  therefore  represents  a  line  through  the  origin. 

3.  Find  the  slope  of  the  lines : 

(a)  x  +  y  =  0.  (c)  3x_-  j\y  =  0. 

(b)  x-y  =  0.  (d)   V2 X  +  y  =  0, 

4.  Draw  a  line  to  represent  Hooke's  Law  E  =  kt,  if  A;  =  10  (see  Ex.  7, 
p.  14),  Let  (be  represented  as  horizontal  lengths  (as  is  x  in  §  23)  and  let 
E  be  represented  by  vertical  lengths  (as  is  y  in  §  23). 

6.  Draw  a  line  to  represent  the  relation  c  =  2  Trr,  where  c  means  the 
circumference  and  r  the  radius  of  a  circle. 

6.  The  number  of  yards  y  in  a  given  length  varies  as  the  number  of 
feet  /  in  the  same  length ;  in  particular,  f=3y.  Draw  a  figure  to 
represent  this  relation. 

7.  If  1  in.  =  2.64  cm.,  show  that  c  =  2.64  i,  where  c  is  the  number  of 
centimeters  and  i  is  the  number  of  inches  in  the  same  length.  Draw  a 
figure. 


II,  §  26] 


,THE  STRAIGHT  LINE 


25 


25.  Slope  Form.  Finally,  consider  a  line  that  does  not  pass 
through  the  origin  and  is  not  parallel  to  either  of  the  axes  of 
coordinates  (Fig.  24)  ;  let  it  intersect  the  axes  Ox,  Oy  at  A, 
B,  respectively,  and  let  P  (x,  y)  be  any  other  point  on  it.  The 
figure  shows  that  the  slope  m  of 
the  line,  i.e.  the  tangent  of  the  . 
angle  a  at  which  the  line  is  in- 
clined  to  the  axis  Ox,  is 


m 


tana  = 


RP 


BR' 
or,  since  RP=:QP-QR=  QP- 

'   _2/ 


Fig.  24 


m 


OB=y. 
-b 


b3ind  BR=OQ=x 


y  =  mx-\-  b„ 


that  is, 

(2) 

where  h  —  OB  is  called  the  intercept  made  by  the  line  on  the 

axis  Oy,  or  briefly  thejji^intercept. 

The  slope  angle  a  at  which  the  line  is  inclined  to  the  axis  Ox 
is  always  understood  as  the  smallest  angle  through  which  the 
positive  half  of  the  axis  Ox  must  be  turned  counterclockwise 
about  the  origin  to  become  parallel  to  the  line. 

26.  Equation  of  a  Line.  On  the  line  AB  of  Fig.  24  take 
any  other  point  P'  -,  let  its  coordinates  be  x',  y',  and  show  that 
y'  =  nnx'  +  b. 

Take  the  point  P*  (x/  y')  outside  the  line  AB  and  show  that 
the  equation  y  =  mx  -f  6  is  not  satisfied  by  the  coordinates  x', 
y'  of  such  a  point. 

For  these  reasons  the  equation  y  =  mx  +  &  is  said  to  represent 
the  line  whose  y4ntercept  is  b  and  whose  slope  is  m;  it  is  also 
called  the  equation  of  this  line.  The  ^/-intercept  OB  =  b  and 
the  slope  m  =  tan  a  together  fully  determine  the  line. 


26  PLANE  ANALYTIC  GEOMETRY  [II,  §  26 

Every  line  of  the  plane  can  he  represented  by  an  equation  of  the 

form 

y  =  mx  +  b, 

excepting  the  lines  parallel  to  the  axis  Oy.  When  the  line  be- 
comes parallel  to  the  axis  Oy,  both  its  slope  m  and  its  y-inter- 
cept  b  become  infinite.  We  have  seen  in  §  22  that  the  equa- 
tion of  a  line  parallel  to  the  axis  Oy  is  of  the  form  x  —  a. 

Reduce  the  equation  3x—2y=5  to  the  form  y  =  mx  +  b  and 
sketch  the  line. 

EXERCISES 

1.  Sketch  the  lines  whose  y-intercept  is  6  =  2  and  whose  slopes  are 
in  =  i,  3,  0,  —  f  ;  write  down  their  equations.         v 

2.  Sketch  the  lines  whose  slope  is  m  =  4/3  and  whose  y-intercepts  are 
0,  1,  2,  5,  —  1,  —  2,  —  6,  —  12.2,  and  write  down  their  equations. 

3.  Sketch  the  lines  whose  equations  are : 

(a)  y=2x+S.        (c)  y=x-^.    (e)  x-y=l.  (gr)  7a;-y+12=0. 

(6)  y=-ia;+L     (d)  x-^y  =  l.     (/)  x-2y-\-2=0.    (h)  4a;+3y+5=0. 

4.  Do  the  points  (1,  5),  (-2,  -1),  (3,  7)  lie  on  the  hne  y=2x-\-S  ? 

5.  A  cistern  that  already  contained  300  gallons  of  water  is  filled  at  the 
rate  of  100  gallons  per  hour.  Show  that  the  amount  A  of  water  in  the 
cistern  n  hours  after  filling  begins  is  ^  =  100  n +300.  Draw  a  figure  to 
represent  this  relation,  plotting  the  values  of  A  vertically,  with  1  vertical 
space  =  100  gallons. 

6.  In  experiments  with  a  pulley  block,  the  pull  p  in  lbs.,  required  to 
lift  a  load  I  in  lbs.,  was  found  to  be  expressed  by  the  equation  |>=.  16 1-\-2. 
Draw  this  line.  How  much  pull  is  required  to  operate  the  pulley  with  no 
load  (i.e.  when  Z  =  0)  ? 

7.  The  readings  of  a  gas  meter  being  tested,  T,  were  found  in  compari- 
son with  those  of  a  standard  gas  meter  S,  and  the  two  readings  satisfied 
the  equation  T  =  SOO  +  1.2  S.  Draw  a  figure.  What  was  the  reading 
T  when  the  reading  S  was  zero  ?  What  is  the  meaning  of  the  slope  of 
the  line  in  the  figure  ? 


II,  §27]  THE  STRAIGHT  LINE  27 

27.   Parallel  and  Perpendicular  Lines.     Two  lines 
y  =  m-ipc  4-  6i ,   y  =  m<^  +  &2 
are  obviously  parallel  if  they  have  the  same  slope,  i.e.  if 

(3)  mi  =  m^. 

Two  lines  y  =  m-^x -\-bi,  y  =  m^  +  h^  are  perpendicular  if  the 
slope  of  one  is  equal  to  minus  the  reciprocal  of  the  slope  of 
the  other,  i.e.  if 

(4)  mYm2  =  —  1. 

For  if  mj  =  tan  «! ,  mg  =  tan  «£ ,  the  condition  that  m^m2  =  —  1 
gives  tan  a2  =  —  1/tan  Oi  =  —  cot «! ,  whence  otg  =  «i  +  i  t. 

EXERCISES 

1.  Write  down  the  equation  of  any  line  :  (a)  parallel  to  y  =  3  ic  —  2, 
(6)  perpendicular  to  y  =  3  x  —  2. 

2.  Show  that  the  parallel  to  y  =  Zx  —  2  through  the  origin  is  y  =  3 x. 

3.  Show  that  the  perpendicular  to  y  =  3  x  —  2  through  the  origin  is 

y=-\x. 

4.  For  what  value  of  h  does  the  line  y  =  3  a;  +  &  pass  through  the 
point  (4,  1)  ?    Find  the  parallel  to  ?/  =  3  x  —  2  through  the  point  (4,  1). 

6.   Find  the  parallel  to  y  =  5  a;  +  1  through  the  point  (2,  3). 

6.  Find  the  perpendicular  to  y  =  2x  —  \  through  the  point  (1,  4). 

7.  What  is  the  geometrical  meaning  of  6i  =  62  in  the  equations 

y  =  m\x  +  61 ,  y  =  mix  +  &2  ? 

8.  Two  water  meters  are  attached  to  the  same  water  pipe  and  the  water 
is  allowed  to  flow  steadily  through  the  pipe.  The  readings  B\  and  Bi  of  the 
two  meters  are  found  to  be  connected  with  the  time  t  by  means  of  the 
equations  i?i  =  2.5f,  i?2  =  2.5«  +  150, 

where  i?i  and  ^2  are  measured  in  cubic  feet  and  t  is  measured  in  seconds. 
Show  that  the  Unes  that  represent  these  equations  are  parallel.  What 
is  the  meaning  of  this  fact  ? 

9.  The  equations  connecting  the  pull  p  required  to  lift  a  load  w  is 
found  for  two  pulley  blocks  to  be 

Pi  =  .05w>  4-  2,  Pi  =  .05w  +  1.5 
Show  that  the  hnes  representing  these  equations  are  parallel.    Explain. 


28        PLANE  ANALYTIC  GEOMETRY    [II,  §  27 

10.  The  equations  connecting  the  pull  p  required  to  lift  a  load  w  is 
found  for  two  pulley  blocks  to  be 

Pi  =  .15w  +  1.5,  p2  =  .05w>  +  1.5. 

Show  that  the  lines  representing  these  equations  are  not  parallel,  but 
that  the  values  of  pi  and  p2  are  equal  when  lo  =  0.     Explain. 

28.  Linear  Function.  The  equation  y  =  mx-\-b,  when  m 
and  b  are  given,  assigns  to  every  value  of  x  one  and  only  one 
definite  value  of  y.  This  is  often  expressed  by  saying  that 
mx  -|-  6  is  a  function  of  x ;  and  as  the  expression  mx  +  5  is  of 
the  first  degree  in  Xj  it  is  called  Si  function  of  the  first  degree  or, 
owing  to  its  geometrical  meaning,  a  linear  function  of  x. 

Examples  of  functions  of  x  that  are  not  linear  are  3  a*  —  5, 
aa^  +  bx-^Cf  x{x  —  l),  1/a;,  sin  a;,  10*,  etc.  The  equations 
y  =  30^  —  5,  y  =  ax^ -\- bx -{•  c,  etc.,  represent,  as  we  shall  see 
later,  not  straight  lines  but  curves. 

The  linear  function  y  =  nix  -h  b,  being  the  most  simple  kind 
of  function,  occurs  very  often  in  the  applications.  Notice  that 
the  constant  b  is  the  value  of  the  function  for  x  =  0.  The  con- 
stant m  is  the  rate  of  change  of  y  with  respect  to  x. 

29.  Illustrations.  Example  1.  A  man,  on  a  certain  date, 
has  $  10  in  bank ;  he  deposits  $  3  at  the  end  of  every  week ; 
how  much  has  he  in  bank  x  weeks  after  date  ? 

Denoting  by  y  the  number  of  dollars  in  bank,  we  have 

2^  =  3a;  +  10. 

His  deposit  at  any  time  a;  is  a  linear  function  of  x.  Notice 
that  the  coefficient  of  x  gives  the  rate  of  increase  of  this  de- 
posit ;  in  the  graph  this  is  the  slope  of  the  line. 

Example  2.  Water  freezes  at  0°  C.  and  32°  F. ;  it  boils  at 
100°  C.  and  at  212°  F. ;  assuming  that  mercury  expands  uni- 
formly, i.e.  proportionally  to  the  temperature,  and  denoting 


II,  §  29]  THE  STRAIGHT  LINE  29 

by  X  any  temperature  in  Centigrade  degrees,  by  y  the  same 
temperature  in  Fahrenheit  degrees,  we  have 

y-32_212-32_9    .      „_8j,,32 

If  the  line  represented  by  this  equation  be  drawn  accurately, 
on  a  sufficiently  large  scale,  it  could  be  used  to  convert  centi- 
grade temperature  into  Fahrenheit  temperature,  and  vice  versa. 

Example  3.  A  rubber  band,  1  ft.  long,  is  found  to  stretch 
1  in.  by  a  suspended  mass  of  1  lb.  Let  the  suspended  mass 
be  increased  by  1  oz.,  2  oz.,  etc.,  and  let  the  corresponding 
lengths  of  the  band  be  measured.  Plotting  the  masses  as  ab- 
scissas and  the  lengths  of  the  band  as  ordinates,  it  will  be 
found  that  the  points  (x,  y)  lie  very  nearly  on  a  straight  line 
whose  equation  is  y  =  ^^x-\-l.  The  experimental  fact  that 
the  points  lie  on  a  straight  line,  i.e.  that  the  function  is  linear, 
means  that  the  extension,  2/  —  1,  is  proportional  to  the  tension, 
i.e.  to  the  weight  of  the  suspended  mass  x  (Hooke's  Law). 

Notice  that  only  the  part  of  the  line  in  the  first  quadrant, 
and  indeed  only  a  portion  of  this,  has  a  physical  meaning. 
Can  this  range  be  extended  by  using  a  spiral  steel  spring  ? 

Example  4.  When  a  point  P  moves  along  a  line  so  as  to 
describe  always  equal  spaces  in  equal  times,  its  motion  is  called 
uniform.  The  spaces  passed  over  are  then  proportional  to  the 
times  in  which  they  are  described,  and  the  coefficient  of  pro- 
portionality, i.e.  the  ratio  of  the  distance  to  the  time,  is  called 
the  velocity  v  of  the  uniform  motion.  If  at  the  time  ^  =  0  the 
moving  point  is  at  the  distance  Sq,  and  at  the  time  t  at  the  dis- 
tance s,  from  the  origin,  then 

S  =  So  +  vt. 
Thus,  in  uniform  motion,  the  distance  s  is  a  linear  function  of 
the  time  t,  and  the  coefficient  of  t  is  the  speed :  v=  (s—  s^/t. 


30  PLANE  ANALYTIC  GEOMETRY  [II,  §  29 

EXERCISES 

1.  If  the  constants  m  and  b  (§28)  are  given  numerically,  any  number 
of  points  of  the  line  can  be  located  by  arbitrarily  assigning  to  the  ab- 
scissa X  any  series  of  values  and  computing  from  the  function  the  corre- 
sponding values  of  the  ordinates.  This  process  is  known  as  plotting  a 
line  by  points.     Two  points  are  sufficient  to  determine  the  line. 

Plot  by  points  the  following  functions : 

(a)  y  =  ^x,  {b)y  =  2x-5,  (c)2/=-3x  +  6, 

(d)y=-^x-i,  (e)y  =  x(ix-l),  {f)y  =  x\ 

{g)y  =  x\  ih)y  =  2'. 

2.  Draw  the  line  represented  by  the  equation  y  =  fx  +  32  of  Ex- 
ample 2,  §  29.  What  is  its  slope  ?  What  is  the  y-intercept  ?  What  is 
the  meaning  of  each  of  these  quantities  if  y  and  x  represent  the  tempera- 
tures in  Fahrenheit  and  in  Centigrade  measure,  respectively  ? 

3.  Represent  the  equation  y  =  ^^  x  +  1  of  Example  3,  §  29,  by  a  figure. 
What  is  the  meaning  of  the  y-intercept  ? 

4.  Draw  the  line  8  =  SQ  +  vt  ot  Example  4,  §  29,  for  the  values  Sq  =  10, 
V  =  3.  What  is  the  meaning  of  t)  ?  Show  that  the  speed  v  may  be 
thought  of  as  the  rate  of  increase  of  s  per  second. 

5.  If,  in  the  preceding  exercise,  v  be  given  a  value  greater  than  3, 
how  does  the  new  line  compare  with  the  one  just  drawn  ? 

6.  If,  in  Ex.  4,  v  is  given  the  value  3,  and  Sq  several  different  values, 
show  that  the  lines  represented  by  the  equation  are  parallel.     Explain. 

7.  In  experiments  on  the  temperatures  at  various  depths  in  a  mine, 
the  temperature  (Centigrade)  T  was  found  to  be  connected  with  the 
depth  d  by  the  equation  r=G0  +  .01^,  where  d  is  measured  in  feet. 
Draw  a  figure  to  represent  this  equation.  Show  that  the  rate  of  increase 
of  the  temperature  was  1°  per  hundred  feet. 

8.  In  experiments  on  a  pulley  block,  the  pull  p  (in  lb.)  required  to 
lift  a  weight  w  (in  lb.)  was  found  to  be  j9  =  .03  w>  +  0.5.  Show  that  the 
rate  of  increase  of  j3  is  8  lb.  per  hundredweight  increase  in  w. 

9.  The  velocity  tj  of  a  body  falling  from  rest  is  proportional  to  the 
time:  V  =  gt,  where.gr  is  a  constant  (about  32  in  English  units).  If  the 
body  is  thrown  down  with  an  initial  velocity  t?o,  the  velocity  at  any  time 


tis 


V  =  Vq-{- 


II,  §30]  THE  STRAIGHT  LINE  31 

Draw  a  figure  to  represent  this  equation  for  g  =  32,  v^  =  10.  Show 
that  g  is  the  rate  of  increase  of  the  velocity  (called  the  acceleration), 

30.    General  Linear  Equation.     The  equation 

in  which.  Ay  B,  C  are  any  real  numbers,  is  called  the  general 
equation  of  the  first  degree  in  x  and  y.  The  coefficients  A,  B,  G 
are  called  the  constants  of  the  equation ;  x,  y  are  called  the 
variables.  It  is  assumed  that  A  and  B  are  not  both  zero. 
The  terms  Ax  and  By  are  of  the  first  degree ;  the  term  C  is 
said  to  be  of  degree  zero  because  it  might  be  written  in  the 
form  Csfi ;  this  term  C  is  also  called  the  constant  term. 

Every  equation  of  the  first  degree, 
(5)  Ax-{^By  ^  C=0, 

in  which  A  and  B  are  not  both  zero,  represents  a  straight  line; 
and  conversely,  every  straight  line  can  be  represented  by  such  an 
equation.  For  this  reason,  every  equation  of  the  first  degree 
is  called  a  linear  equation. 

The  first  part  of  this  fundamental  proposition  follows  from 
the  fact  that,  when  B  is  not  equal  to  zero,  the  equation  can  be 
reduced  to  the  form  y  =  mx  +  6  by  dividing  both  sides  by  B ; 
and  we  know  that  y  =  mx  +  b  represents  a  line  (§  25).  When 
B  is  equal  to  zero,  the  equation  reduces  to  the  form  x  =  a^ 
which  also  represents  a  line  (§  22). 

The  second  part  of  the  theorem  follows  from  the  fact  that 
the  equations  which  we  have  found  in  the  preceding  articles 
for  any  line  are  all  particular  cases  of  the  equation 
Ax -\- By +0=0. 

This  equation  still  expresses  the  same  relation  between  x 
and  y  when  multiplied  by  any  constant  factor,  not  zero.  Thus, 
any  one  of  the  constants  A,  B,  C,  if  not  zero,  can  be  reduced 
to  1  by  dividing  both  sides  of  the  equation  by  this  constant. 


32 


PLANE  ANALYTIC  GEOMETRY  [II,  §  30 


The  equation  is  therefore  said  to  contain  only  two  (not  three) 
essential  constants. 

31.   Conditions  for  Parallelism  and  for  Perpendicularity. 

It  is  easy  to  recognize  whether  two  lines  whose  equations  are 
Ax  +  By -\-C=0  and  A'x -f- B'y  +  C"  =  0  are  parallel  or  per- 
pendicular. The  lines  are  parallel  if  they  have  the  same  slope7\ 
and  they  are  perpendicular  (§  27)  if  the  product  of  their  slopes 
is  equal  to  —1.  The  slopes  of  our  lines  are  —  A/B  and 
—  A/B' ;  hence  these  lines  are  parallel  if  —  A/B  =  —  A!/Bf, 
i'e.  if  A.B  =  A''.B'', 

and  they  are  perpendicular  if 


A    A 
b'  B 


,  =  -1^ 


i.e.  if 


AA'  -hBB'  =  0. 


32.  Intercept  Form.  If  the  constant  term  C  in  a  linear 
equation  is  zero,  the  equation  represents  a  line  through  the 
origin.  For,  the  coordinates  (0,  0)  of  the  origin  satisfy  the 
equation  Ax  +  By  =  0. 

If  the  constant  terra  C  is  not  equal  to  zero,  the  equation 
Ax -{- By  -\-  C=0  can  be  divided  by  (7;  it  then  reduces  to  the 

form  A     ,  B     ,  ^       n 

—X  H — v  +  1  =0. 
C       0 

If  A  and  B  are  both  different  from  zero,  this  can  be  written : 


-C/A'   -  C/B 
or  putting  —  C/A  =  a,  —  C/B  =  b : 

(6) 


a      o 


The  conditions  A=^0,  B  =^  0  mean  fig.  25 

evidently  that  the  line  is  not  parallel  to  either  of  the  axes. 
Therefore  the  equation  of  any  line,  not  passing  through  the 


II,  §32]  THE  STRAIGHT  LINE  33 

origin,  and  not  parallel  to  either  axis,  can  be  written  in  tlie 
form  (6).  With  y  =  0  this  equation  gives  x=:  a;  with  x  =  0 
it  gives  y  =  b.     Thus 

are  the  intercepts  (Fig.  25)  made  by  the  line  on  the  axes  Ox, 
Oy,  respectively  (see  §  25). 

EXERCISES 

1.  Write  down  the  equations  of  the  line  whose .  intercepts  on  the 
axes  Ox,  Oy  are  5  and  —  3,  respectively  ;  the  line  whose  intercepts  are 

—  I  and  7  ;  the  line  whose  intercepts  are  —  1  and  —  |.  Sketch  each  of 
the  lines  and  reduce  each  of  the  equations  to  the  form  Ax+By-\-  (7=0,  so 
that  A,  B,  C  are  integers. 

2.  Find  the  intercepts  of  the  lines :  Sx  —  2y  =  l,x-{-Ty-\-l=0, 

—  Sx+ly  —  5  =  0.  Try  to  read  off  the  values  of  the  intercepts  directly 
from  these  equations  as  they  stand. 

3.  In  Ex.  2,  find  the  slopes  of  the  lines. 

4.  Prove  (6),  §  32  by  equality  of  areas,  after  clearing  of  fractions. 
6.    What  is  the  equation  of  the  axis  Oy  ?  of  the  axis  Ox  ? 

6.  What  is  the  value  of  B  such  that  the  line  represented  by  the  equa- 
tion 4:X  +  By  —  li  =  0  passes  through  the  point  (—6,  17). 

7.  What  is  the  value  of  A  such  that  the  line  Ax  i-l  y=10  has  its 
oj-intercept  equal  to  —  8  ? 

8.  Reduce  each  of  the  following  equations  to  the  intercept  form  (6) , 
and  draw  the  lines  : 

(a)  3  X  -  5  ?/  -  16  =  0.  (b)  X -\- ^  y -\- 1  =  0. 

4x-_3j^^-6^2  (^d)  5x=:Sx  +  y-10. 

9.  Reduce  the  equations  of  Ex.  8  to  the  slope  form  (2),  §  25. 

10.   Find  the  equation  of  the  line  of  slope  6  passing  through  the  point 
(6,  -  5). 

D 


34  PLANE  ANALYTIC  GEOMETRY  [II,  §  32 

11.  Show  that  the  points  (-  1,  -  7),  (i,  -  3),  (2,  2),  (-2,  -  10) 
lie  on  the  same  line, 

12.  Find  the  area  of  the  triangle  formed  by  the  lines  x  -\-  y  =  0, 
x  —  y  =  OfX  —  a  =  0. 

13.  Show  that  the  line  4(aj  —  a)  -f-  5(y  —  6)  =  0  is  perpendicular  to  the 
line  5x  —  4y  —  10  =  0  and  passes  through  the  point  (a,  b). 

14.  A  line  has  equal  positive  intercepts  and  passes  through  (—5,  14). 
What  is  its  equation  ?  its  slope  ? 

16.  If  a  line  through  the  point  (6,  7)  has  the  slope  4,  what  is  its 
y-intercept  ?  its  x-intercept  ? 

16.  The  Reaumur  thermometer  is  graduated  so  that  water  freezes  at 
O'^  and  boils  at  SO*^-  Draw  the  line  that  represents  the  reading  i?  of  the 
Reaumur  thermometer  as  a  function  of  the  corresponding  reading  C  of 
the  Centigrade  thermometer. 

17.  Express  the  value  of  a  note  of  $  1000  at  the  end  of  the  first  year  as 
a  function  of  the  rate  of  interest.  At  6  %  simple  interest  its  value  is  what 
function  of  the  time  in  years  ? 

33.  Line  through  One  Point.  To  find  the  line  of  given 
slope  7%  through  a  given  point  Pi{xi,  yi),  observe  that  the 
equation  must  be  of  the  form  (2),  viz. 

since  this  line  has  the  slope  m^.  If  this  line  is  to  pass  through 
the  given  point,  the  coordinates  ajj,  yi  must  satisfy  this  equa- 
tion, i.e.  we  must  have 

Vi  =  iriiXi  +  h. 

This  equation  determines  b,  and  the  value  of  b  so  found  might 
be  substituted  in  the  preceding  equation.  But  we  can  eliminate 
b  more  readily  between  the  two  equations  by  subtracting  the 
latter  from  the  former.     This  gives 

as  the  equation  of  the  line  of  slope  mj  through  Pi(x^,  y^. 


II,  §  34] 


THE  STRAIGHT  LINE 


35 


The  problem  of  finding  a  line  through  a  given  point  parallel^ 
or  perpendicular,  to  a  given  line  is  merely  a  particular  case  of 
the  problem  just  solved,  since  the  slope  of  the  required  line  can 
be  found  from  the  equation  of  the  given  line  (§  27).  If  the 
slope  of  the  given  line  is  m^  =  tan  a^,  the  slope  of  any  parallel 
line  is  also  m^,  and  the  slope  of  any  line  perpendicular  to  it  is 

1 


mg  =  tan  (a^  +  ^tt)  =  —  cot  Wj  =  — 


m, 


34.  Line  through  Two  Points,  To  find  the  line  through  two 
given  points,  Pi{x^,  Pi),  P^ix^,  y^,  observe  (Fig.  26)  that  the 
slope  of  the  required  line  is  evi- 
dently 


m,  = 


_y2-yi_^y 


Xa    Xi 


Ax 


if,  as  in   §  9,  we  denote  by  Ax,  Ay 

the  projections  of  P^Pz  on  Ox,  Oy; 

and  as  the  line  is  to  pass  through  (x^,  y{),  we  find  its  equation 

by  §  33  as 


y-yi  = 


y2-yi 

Xt>  —  3/1 


or 


y-yi  =  ^{x 

Ax 


(x  -  X,), 


0. 


The  equation  of  the  line  through  two  given  points  (xi,  yi), 
(^2j  2/2)  can  also  be  written  in  the  determinant  form 

X     y      1 

xi    yi    1  =0, 

^2    2/2     1 
which  (§  14)  means  that  the  point  (x,  y)  is  such  as  to  form 
with  the  given  points  a  triangle  of  zero  area.     By  expanding 
the  determinant  it  can  be  shown  that  this  equation  agrees  with 
the  preceding  equation. 


36  PLANE  ANALYTIC  GEOMETRY  [II,  §  34 

EXERCISES 

1.  Find  the  equation  of  the  line  through  the  point  (—7,  2)  parallel  to 
the  line  y  =  Sx. 

2.  Show  that  the  points  (4,  —  3),  (—  5,  2),  (5,  20)  are  the  vertices  of 
a  right  triangle. 

3.  Find  the  equation  of  the  line  through  the  point  (—6,  —3)  which 
makes  an  angle  of  30°  with  the  axis  Ox ;  30°  with  the  axis  Oy. 

4.  Does  the  line  of  slope  |  through  the  point  (4,  3)  pass  through  the 
point  (-5,  -4)  ? 

6.  Find  the  equation  of  the  line  through  the  point  (—  2, 1)  parallel  to 
the  line  through  the  points  (4,  2)  and  (—3,  —  2). 

6.  Find  the  equations  of  the  lines  through  the  origin  which  trisect 
that  portion  of  the  line  6  x  —  6  y  =  60  which  lies  in  the  fourth  quadrant. 

7.  What  are  the  intercepts  of  the  line  through  the  points  (2,  —  3), 
C-5,4)? 

8.  Show  that  the  equation  of  the  line  through  the  point  (a,  b)  per- 
pendicular to  the  line  Ax  +  By  -}-  C  =  0  ia  (x  —  a) /A  =  (y  —  bys. 

9.  Find  the  equations  of  the  diagonals  of  the  rectangle  formed  by  the 
lines  x-{-a  =  0,  x  —  b  =  0,  y  +  c  =  0,  y  —  d  =  0. 

10.  Find  the  equation  of  the  perpendicular  bisector  of  the  line  joining 
the  points  (4,  —  5)  and  (—3,  2).  Show  that  any  point  on  it  is  equally 
distant  from  each  of  the  two  given  points. 

11.  Find  the  equation  of  the  line  perpendicular  to  the  line  Ax  —  Sy-\-G=0 
that  passes  through  the  midpoint  of  (—  4,  7)  and  (2,  2). 

12.  What  are  the  coordinates  of  a  point  equidistant  from  the  points 
(2,  —  3)  and  (—  5,  0)  and  such  that  the  line  joining  the  point  to  the  origin 
has  a  slope  1  ? 

13.  In  an  experiment  with  a  pulley-block  it  is  assumed  that  the  rela- 
tion between  the  load  I  and  the  pull  p  required  to  lift  it  is  linear.  Find 
the  relation  if  p  =  8  when  I  =  100,  and  p  =  12  when  I  =  200. 

14.  In  an  experiment  in  stretching  a  brass  wire  it  is  assumed  that  the 
elongation  E  is  connected  with  the  tension  t  by  means  of  a  linear  relation. 
Find  this  relation  if  i  =  18  lb.  when  E  =  A  in.,  and  <  =  58  lb.  when 
^  =  .3  in. 


CHAPTER  III 
RELATIONS   BETWEEN   TWO   OR   MORE   LINES 

35.  Intersection  of  Two  Lines.  The  point  of  intersection 
of  any  two  lines  is  found  by  solving  the  equations  of  the  lines  as 
simultaneous  equations.  For,  the  coordinates  of  the  point  of 
intersection  must  satisfy  each  of  the  two  equations,  since  this 
point  lies  on  each  of  the  two  lines ;  and  it  is  the  only  point 
having  this  property.     Thus,  by  solving  the  equations 

3  a; +  5^-34  =  0, 

we  find  X  =  S,  y  =  5]  hence  (3,  5)  is  the  point  of  intersection 
of  the  two  lines  represented  by  these  equations. 

36.  Particular  Cases.  The  equations  of  any  two  lines  being 
given,  say 

(1)  aix  +  hy  =  fci, 
a^x  -f-  622/  =  ^2) 

we  find  by  the  usual  method,  that  is,  first  multiplying  by  62?  &i 
and  subtracting,  then  multiplying  by  02,  ai  and  subtracting : 

(2)  ,  (ai&2  —  a2bi)x  =  fci&2  —  ^2^) 

{a^2  —  <^^i)y  =  GbJ^2  —  Ct2^1« 
The  expression  aih^  —  a^bi  is  called  the  determinant  of  the 
equations.     Two  cases  must  be  distinguished  according  as  this 
determinant  is  :?i:  0  or  =  0. 

(a)  If  a^2  —  <^2^i  =9^  ^>  which  means  by  §  31  that  the  lines  are 
not  parallel,  we  can  divide  the  equations  (2)  by  this  determi- 
nant and  thus  find  x  and  y.    If,  in  particular,  ki  and  k2  are  both 

37 


38  PLANE  ANALYTIC  GEOMETRY         [III,  §  36 

zero,  tiiat  is,  if  the  equations  (1)  are  homogeneous  and  hence 

represent  two  lines  through  the  origin,  we  find  from  (2)  a;  =  0 

and  2/  =  0,  as  was  to  be  expected. 

(6)  If  ai&2  —  ct2^i  =  0,  that  is,  if  the  lines  (1)  are  parallel,  we 

cannot  in  (2)  divide  by  0162  —  (h^i ;   the  equations  (2)  then 

become 

0'X  =  ^162  —  k2bif 

0  . 2/  =  aik2  —  02^1, 
and  cannot  be  satisfied  by  any  values  of  x  and  y  unless  the 
right-hand  members  are  both  zero.     In  the  latter  case  we  have 

rt'2        ^        ^2 

that  is,  the  second  equation  is  merely  a  multiple  of  the  first. 
In  this  case  the  two  equations  (1)  represent  the  same  line  and 
have  therefore  all  points  in  common. 

EXERCISES 

1.  Find  the  coordinates  of  the  points  of  intersection  of  the  following 
lines  ;  and  check  by  a  sketch  : 

(a')  J5x-7  2/  +  ll=0,  ,^.  f  3x-}-2y=0,  ,.  f  2.4a;+3.1  y=  4.6, 
^^    [3ic+2y-12=0.      ^^    [6x-4y+4=0.     ^^     [     .8x  +  2y  =  6.2. 

2.  Do  the  following  pairs  of  lines  intersect,  or  are  they  parallel  or 
coincident  ? 

.  .     f3a;-6y-8=0,     ...     f2x-6y-4=0,     ,  .     f    x  +  iy  =  0, 
^^    I     x-2y+l  =  0.     ^^    [    x-3y-2=0.      ^^    |2x  +  3y  =  0. 

3.  Show  that  the  condition  that  the  three  lines  Ax -{-  By  +  (7=0, 
A'x  4-  B'y  +  C"  =  0,  A"x  +  B"y  +  C"  =  0  meet  at  a  point  is 

ABC 

A'      B'      C   =0. 

A^-     B"     C" 

4.  Show   that  the    straight  lines  3x  +  y  —  1=0,  jc-3y  +  13  =  0, 
2x  —  y  -^6  =  0  have  a  common  point. 


Ill,  §  37]  TWO  OR  MORE  LINES  39 

6.  Show  that  the  lines  joining  the  midpoints  of  the  sides  of  any  tri- 
angle divide  the  triangle  into  four  equal  triangles. 

6.  Show  that  the  altitudes  of  any  triangle  meet  in  a  point. 

7.  Show  that  the  medians  of  any  triangle  meet  in  a  point. 

8.  Show  that  the  line  through  the  origin  perpendicular  to  the  line 
through  the  points  (a,  0)  and  (0,  b)  meets  the  lines  through  the  points 
(a,  0\,  {—b,  b)  and  (0,  &),  (a,  -va)  in  a  common  point. 

Xejt.  &XX  M^,  y-o  -^ 

37.  Angle  between  Two  Lines.  We  shall  understand  by 
the  angle  (/,  l')=  6  between  two  lines  I  and  V  the  least  angle 
through  which  I  must  be  turned  coun- 
terclockwise about  the  point  of  inter- 
section to  come  to  coincidence  with  V. 
This  angle  0  is  equal  to  the  differ- 
ence of  the  slope  angles  a,  a'  (Fig.  27) 
of  the  two  lines.  Thus,  if  a'  >  a,  we 
have  6  =  a'  —  a,  since  a'  is  the  exterior  Fig.  27 

angle  of  a  triangle,  two  of  whose  interior  angles  are  a  and  6, 

It  follows  that 

/o\  i.      /I      i.      /  f        \       tan  a'  —  tana. 

(3)  tan  6  =  tan  (a'  —  a)  = • 

^  ^  ^  ^     1  H-  tan  a  tan  a' 

If  the  equations  of  I  and  V  are 

y  =  mx  -^  b,  y  =  m'x  -|-  6', 
respectively,  we  have  tan  a  =  m,  tan  «'  =  m' ;  hence 

(4)  t^ne^^^^^^^r 
^  ^  1  -f  mm' 

If  the  equations  of  I  and  Z'  are 

Ax -\- By +  0=0, 
A'x  +  B'y  +  C"  =  0, 
respectively,  we  have  tan  a  =  —  A/B,  tan  a'  =  —  A'/B' ;  hence 

AB'  -  A'B 


(5)  tan  6  = 


AA!  +  BB' 


40  PLANE  ANALYTIC  GEOMETRY         [III,  §  38 

38.  It  follows,  in  particular,  that  the  two  lines  I  and  V,  §  37, 
are  parallel  if  and  only  if 

m'  =  m,      or  AB'  -  A'B  =  0 ; 
and  they  are  perpendicular  to  each  other  if  and  only  if  "^ 

m'  =  --,ovAA'  +  BB'  =0. 
m 

(Compare  §§27,  31.)     Hence,  to  write  down  the  equation  of 

a  line  parallel  to  a  given  line,  replace  the  constant  term  by  an 

arbitrary  constant ;  to  write  down  the  equation  of  a  line  per- 

pendicular  to  a  given  line,  interchange  the  coefficients  of  x  and 

y,  changing  the  sign  of  one  of  them,  and  replace  the  constant 

term  by  an  arbitrary  constant. 

EXERCISES 

1.  Determine  whether  the  following  pairs  of  lines  are  parallel  or  per- 
pendicular :  Sx  +  2y-Q  =  0,  2a;-3y  +  4  =  0;  6a;  +  3y-6  =  0, 
10x  +  6y  +  2  =  0;2ic-f6y-14=0,  8a;-3y  +  6=0. 

2.  Find  the  point  of  intersection  of  the  line  6a;  +  8y-f-17  =  0  with  its 
X)erpendicular  through  the  origin. 

3.  Find  the  point  of  intersection  of  the  lines  through  the  points  (6,  —2) 
and  (0,  2),  and  (4,  6)  and  (-  1,  -4). 

4.  Find  the  perpendicular  bisector  of  the  line-seginent  joining  the 
point  (3,  4)  to  the  point  of  intersection  of  the  lines  2x  —  y-\-l  =  0  and 
3  x  +  y  -  16  =  0. 

6.  Find  the  lines  through  the  point  of  intersection  of  the  lines  5  x— y =0, 
x-t-7y  —  9  =  0  and  perpendicular  to  them. 

6.  Find  the  area  of  the  triangle  formed  by  the  lines  3  a;  +  4  y  =  8, 
6  a;  —  5  2/  =  30,  and  x  =  0. 

7.  Find  the  area  of  the  triangle  formed  by  the  lines  a;  +  y  —  1  =  0, 
2  X  +  y  +  6  =  0,  and  X  -  2  y  -  10  =  0. 

8.  Find  the  point  of  intersection  of  the  lines 

a     0  ha 


Ill,  §39]  TWO  OR  MORE  LINES  41 

9.    Find  the  area  of  the  triangle  formed  by  the  lines  y  =  m\X  +  6i, 
y  =  mix,  +  62  and  the  axis  Ox. 

10.  The  vertices  of  a  triangle  are  (5,  -  4),  (—  3,  2),  (7,6).     Find  the 
equations  of  the  medians  and  their  point  of  intersection. 

11.  Find  the  angle  between  the  lines  4  x— 3  y— 6=0  and  x—1 1/+6=0. 

12.  Find  the  tangent  of  the  angle  between  the  lines  (a)  4ic— 3y+6=0 
and9a;  +  2y-8  =  0;   (6)  3a;4-6y- 11  =  0  and  a;  +  2y-3  =  0. 

13.  Find  the  two  lines  through  the  point  (6,  10)  inclined  at  45°  to 
the  line  3  a;  -  2  ?/  -  12  =  0. 

14.  Find  the  lines  through  the  point  (—  3,  7)  such  that  the  tangent  of 
the  angle  between  each  of  these  lines  and  the  line  6a;  —  2y  +  ll=0isJ. 

15.  Show  that  the  angle  between  the  lines  Ax  +  By  +  0  =  0  and 

{A  +  B)x-(A-B)y  +  D  =  0  is  45°. 

16.  Find   the  lines    which    make    an    angle   of   45°    with   the  line 
4x  —  7  y  +  6  =0  and  bisect  the  portion  of  it  intercepted  by  the  axes. 

17.  The  hypotenuse  of  an  isosceles  right-angled  triangle  lies  on  the  line 
3a;  —  6y—  17=0.     The  origin  is  one  vertex  ;  what  are  the  others  ? 

39.  Polar  Equation  of  Line.  The  position  of  a  line  in  the 
plane  is  fully  determined  by  the  length  p  =  ON  (Fig.  28)  of  the 
perpendicular  let  fall  from  the  origin  on 
the  line  and  the  angle  ft  =  xOJSf  made  by 
this  perpendicular  with  the  axis  Ox. 

Then  p  and  ft  are  evidently  the  polar 
coordinates  of  the  point  -A''  (§  16).  Let 
P  be  any  point  of  the  line  and  OP  =  r, 
xOP=  <f>  its  polar  coordinates.  As  the 
projection  of  OP  on  the  perpendicular 
ON  is   equal  to  ON,  and  the  angle  NOP  —  <f>  —  ft,  we  have 

(6)  rcos(<f>-ft)=p. 

This  is  the  equation  of  the  line  NP  in  polar  coordinates. 


42  PLANE  ANALYTIC  GEOMETRY         [III,  §  40 

40.  Normal  Form.  The  last  equation  can  be  transformed  to 
Cartesian  coordinates  by  expanding  the  cosine  : 

r  cos  </>  cos  )8  +  r  sin  <^  sin  )3  =p 

and  observing  (§  17)  that  r  cos  </>  =  «,  r  sin  </>  =  ?/ ;  the  equation 
then  becomes 

(7)  xco8p+  y8inp=:p. 

This  equation,  which  is  called  the  normal  form  of  the  equation 
of  the  line,  can  be  read  off  directly  from  the  figure ;  it  means 
that  the  sum  of  the  projections  of  x  and  y  on  the  perpendicular 
to  the  line  is  equal  to  the  projection  of  r  (§  20). 

Observe  that  in  the  normal  form  (7)  the  number  p  is  always 
positive,  being  the  distance  of  the  line  from  the  origin,  or  the 
radius  vector  of  the  point  N.  Hence  x  cos  fi-\-ysin  ft  is  always 
positive ;  this  also  appears  by  considering  that  xcos  (S-^y  sin  /? 
is  the  projection  of  the  radius  vector  OP  on  ON,  and  that  this 
radius  vector  makes  with  ON  an  angle  that  cannot  be  greater 
than  a  right  angle.  , 

The  angle  p  =  xON  is,  as  a  polar  angle  (§  16),  always  under- 
stood to  be  the  angle  through  which  the  axis  Ox  must  be  turned 
counterclockwise  about  the  origin  to  make  it  coincide  with  ON; 
it  can  therefore  have  any  value  from  0  to  2  tt.  By  drawing  the 
parallel  to  the  line  NP  through  the  origin  it  is  readily  seen 
that,  if  a  is  the  slope  angle  of  the  line  NP,  we  have 

fi  =  a-{-^Tr   or  ^  =  a  +  |7r 

according  as  the  line  lies  on  one  side  of  the  origin  or  the  other, 
angles  differing  by  2  tt  being  regarded  as  equivalent.  Thus,  in 
Fig.  28,  a  =  120°,  /5  =  a  +  |  tt  =  120°  +  270°  =  390°,  which  is 
equivalent  to  30°.  For  a  parallel  on  the  opposite  side  of  the 
origin  we  should  have  y8  =  «+  i  tt  =  120°  +  90°  =  210°. 


Ill,  §  41]  TWO  OR  MORE  LINES  43 

41.   Reduction  to  Normal  Form.     The  equation 

Ax-{-By+C=0 
is  in  general  not  of  the  form  (7),  since  in  the  latter  equation 
the  coefficients  of  x  and  y,  being  the  cosine  and  sine  of  an 
angle,  have  the  property  that  the  sum  of  their  squares  is  equal 
to  1,  while  in  the  former  equation  the  sum  of  the  squares  of 
A  and  B  is  in  general  not  equal  to  1.    But  the  general  equation 

Ax  +  By-^C=0 

can  be  reduced  to  the  normal  form  (7)  by  multiplying  it  by 
a  factor  k  properly  chosen ;  we  know  (§  30)  that  the  equation 

kAx  +  kBy  +  kO=0 

represents  the  same  line  as  does  the  equation  Ax-{-By+C=0. 
Now  if  we  select  k  so  that 

kA  =  cos  p,    kB  =  sin  /3,    kC=—p, 
the  equation  Ax -{-  By  -{-  G  =  0  reduces  to  the  normal   form 
X  cos  p  -{-y  sin  y3  —  p  =  0.     The  first  two  conditions  give 

k^A^  +  fc2jB2  ^  cos2  p  +  sin2  yS  =  1, 

whence  A;  =  ± 


■yjA^  +  B^      ^ 

Since  the  right-hand  member  p  in  the  normal  form  (7)  is  posi- 
tive, the  sign  of  the  square  root  must  be  selected  so  that  kC 
becomes  negative.     We  have  therefore  the  rule  : 

To  reduce  the  general  equation  Ax -\- By -\- C =0  to  the  normal 
form 

xcos  p  -\-y  sin  /3  —p=  0, 

divide  by  —  VA^  +  B^  when  C  is  positive  and  by  -{-■\/AF+^ 
when  C  is  negative. 

Then  the  coefficients  of  x  and  y  will  be  cos  ft  sin  /3,  respec- 
tively, and  the  constant  term  will  be  the  distance  p  of  the  line 
from  the  origin. 


44  PLANE  ANALYTIC  GEOMETRY         [III,  §  41 

Thus,  to  reduce  Sx-\-2y-\-5  =  0to  the  normal  form,  divide 
by  -V3M^  =  -Vl3;  this  gives 

cosy8  = =,  smy8  = z=j  —p  = 


Vl3'        '  Vl3'      ^  V13' 


i.e.  the  normal  form  is 
3 


:2/  = 


V13        Vl3        V13 

The  perpendicular  to  the  line  from  the  origin  has  the  length 
6/Vl3 ;  and  as  both  cos  ft  and  sin  ft  are  negative,  this  perpen- 
dicular lies  in  the  third  quadrant.     Draw  the  line. 

Reduce  the  equation  3x-{-2y  —  5  —  0to  the  normal  form. 

42.  Distance  of  a  Point  from  a  Line.  If,  in  Fig.  28,  we 
take  instead  of  a  point  P  on  the  line  any  point  P^  {xi,  y^) 
not  on  the  line  (Fig.  29),  the  expression 
Xi  cos  /3  +  yi  sin  p  is  still  the  projection  on 
ON  (produced  if  necessary)  of  the  radius 
vector  OPi.  But  this  projection  OS  differs 
from  the  normal  ON  =  p  to  the  line.  The 
figure  shows  that  the  difference 

Xi cos p  +  yisin p ^ p  =  OS  —  0N=  NS  fio.  29 

is  equal  to  the  distance  NiPi  of  the  point  P^  from  the  line. 

Thus,  to  find  the  distance  of  any  point  P^  (xi,  yi)  from  a  line 
whose  equation  is  given  in  the  normal  form 
xcos  p  +  ysinp  —  p=  0, 
it  suffices  to  substitute  in  the  left-hand  member  of  this  equa- 
tion for  Xf  y  the  coordinates  x^,  y^  of  the  point  Pj.  The  expression 

Xicos  p  +  yisin  p —p 
then  represents  the  distance  of  P^  from  the  line. 

If  this  expression  is  negative,  the  point  Pi  lies  on  the  same 
side  of  the  line  as  does  the  origin ;  if  it  is  positive,  the  point 


in,  §  43] 


TWO  OR  MORE  LINES 


45 


Pi  lies  on  the  opposite  side  of  the  line.    Any  line  thus  divides  the 
plane  into  two  regions  which  we  may  call  the  positive  and  nega- 
tive regions ;  that  in  which  the  origin  lies  is  the  negative  region. 
To  find  the  distance  of  a  point  P^  (aji,  ^i)  from  a  line  given  in 

the  general  form 

Ax  +  By+C=0, 

we  have  only  to  reduce  the  equation  to  the  normal  form  (§  41) 

and  then  apply  the  rule  given  above.    Thus  the  distance  is 

Ax^  4-  By^  +  O       ^^       Ax^±ByjJ-C_ 

according  as  C  is  positive  or  negative. 

43.    Bisector  of  an  Angle.     To  find  the  bisectors  of  the 
angles  between  two  lines  given  in  the  normal  form 
xGos  ft-\-y  sin  ^8  — p  =  0, 
a;  cos /?'  +  ?/ sin  ^' —  p' =  0, 
observe  that  for  any  point  on  either  bisector  its  distances  from 
the  two  lines  must  be  equal  in  absolute  value.      Hence  the 
equations  of  the  bisectors  are 

xGos ^-{-ysm  (i—p=±(x cos  /3'  -\-y  sin  /8'  —p'). 
To  distinguish  the  two  bisectors,  ob- 
serve that  for  the  bisector  of  that  pair 
of  vertical  angles  which  contains  the 
origin  (Fig.  30)  the  perpendicular  dis- 
tances are,  in  one  angle  both  positive, 
in  the  other  both  negative ;  hence  the 
plus  sign  gives  this  bisector. 

If  the   equations  of  the   lines   are 
given  in  the  general  form 

Ax-^By-^C  =  0,    A'x+B'y-[-C'  =  0, 
first  reduce  the  equations  to  the  normal  form,  and  then  apply 
the  previous  rule. 


N^ 

/     / 

N 

fv 

Xn— 

/''>-' 

\  "n 

~\\ 

1 

\/  1 

^ 

^  1 

^x 

•       0 

/     / 

/ 

1          ' 

Fig.  30 


46  PLANE  ANALYTIC  GEOMETRY         [HI,  §  43 

EXERCISES 

1.  Draw  the  lines  represented  by  the  following  equations  : 
(a)  r  cos  (0  -  ^  ir)  =  6.  (e)   r  cos  (0  +  f  tt)  =  3. 
(6)  r  cos  (0  —  t)  =  4.                        (/)  r  sin  (0  -  ^  tt)  =  8. 

(c)  r  cos  0  =  10.  ($r)  r  sin  (0  +  |  ir)  =  7. 

(d)  r  sin  0  =  5.  (Ji)  r  cos  (0  —  |  tt)  =  0. 

2.  In  polar  coordinates,  find  the  equations  of  the  lines  :  (a)  parallel  to 
and  at  the  distance  6  from  the  polar  axis  (above  and  below)  ;  (6)  per- 
pendicular to  the  polar  axis  and  at  the  distance  4  from  the  pole  (to  the 
right  and  left)  ;  (c)  inclined  at  an  angle  of  ^  tt  to  the  polar  axis  and  at 
the  distance  12  from  the  pole. 

3.  Express  in  polar  coordinates  the  sides  of  the  rectangle  0 ABC  it 
OA  =  6  and  AB  =  9,  OA  being  taken  as  polar  axis. 

4.  What  lines  are  represented  by  (7)  when  p  is  constant,  while  /3 
varies  from  zero  to  2  tt  ?  What  lines  when  p  varies  while  /3  remains  con- 
stant? 

6.  The  perpendicular  from  the  origin  to  a  line  is  5.  units  in  length  and 
makes  an  angle  tan-i  ^  with  the  axis  Oac.  Find  the  equation  of  the  line. 

6.  Reduce  the  equations  of  Ex.  8,  p.  33,  to  the  normal  form  (7). 

7.  Find  the  equations  of  the  lines  whose  slope  angle  is  160°  and  which 
are  at  the  distance  4  from  the  origin. 

8.  What  is  the  equation  of  the  line  through  the  point  ( —  3,  5)  whose 
perpendicular  from  the  origin  makes  an  angle  of  120^  with  the  axis  Ox  ? 

9.  For  the  line  7x  —  24y  —  20=0  find  the  intercepts,  slope,  length 
of  perpendicular  from  the  origin  and  the  sine  and  cosine  of  the  angle 
which  this  perpendicular  makes  with  the  axis  Ox. 

10.  Find  by  means  of  sin  /3  and  cos/3  the  quadrants  crossed  by  the  line 
4ic  — 5y  =  8. 

11.  Put  the  following  equations  in  the  form  (7)  and  thus  find  p,  sin  /3, 

cos  /9: 

(a)y=:mx  +  b.         (6)?+|  =  l.         (c)3a;  =  4y. 
a      0 

IS.  Is  the  point  (3,  —  4)  on  the  positive  or  negative  side  of  the  line 
through  the  points  (—  5,  2)  and  (4,  7)  ? 


Ill,  §  43]  TWO  OR  MORE  LINES  47 

13.  Is  the  point  (—1,  —  f )  on  the  positive  or  negative  side  of  the  line 
4:X-9y-8  =  0? 

14.  Find  by  means  of  an  altitude  and  a  side  the  area  of  the  triangle 
formed  by  the  lines  Zx  +  2y  +  10  =  0,  ^x-3y+lQ  =  0,  2x  +  y-4: 
=  0.     Check  the  result  with  another  altitude  and  side. 

15.  Find  the  distance  between  the  parallel  lines  (a)  Sx— 5y— 4  =  0 
and  6  a;  —  10  y  +  7  =  0  ;  (6)  5  x  +  7  2/  +  9  =  0  and  16  re  +  21  y  —  3  =  0. 

16.  What  is  the  length  of  the  perpendicular  from  the  origin  to  the  line 
through  the  point  (—5,  —  4)  whose  slope  angle  is  60°  ? 

17.  What  are  the  equations  of  the  lines  whose  distances  from  the 
origin  are  6  units  each  and  whose  slopes  are  f  ? 

18.  Find  the  points  on  the  axis  Ox  whose  perpendicular  distances  from 
the  line  24  a:  —  7  t^  —  16  =  0  are  ±5. 

19.  Find  the  point  equidistant  from  the  points  (4,  —  3)  and  (—2,  1), 
and  at  the  distance  4  from  the  line  Sx  —  Ay  —  6  =  0. 

20.  Find  the  line  parallel  tol2a;  —  5y  —  6  =  0  and  at  the  same  distance 
from  the  origin  ;  farther  from  the  origin  by  a  distance  3. 

21.  Find  the  two  lines  through  the  point  (1,  ^^)  such  that  the  perpen- 
diculars let  fall  from  the  point  (6,  5)  are  of  length  5. 

22.  Find  the  line  perpendicular  to  4  a;  —  7  y  —  10  =  0  which  crosses  the 
axis  Ox  at  a  distance  6  from  the  point  (—  2,  0). 

23.  Find  the  bisectors  of  the  angles  between  the  lines:  (a)  x—y  —4=  0 
and  3 x  + 3 2^  + 7  =  0;  (6)  6a:-12y-16  =  0  and  24x  +  7y  +  60  =  0. 

24.  Find  the  bisectors  of  the  angles  of  the  triangle  formed  by  the  lines 
5x  +  12  2/  +  20  =  0,  4x-32/-6  =  0,  3x-4y+5  =  0  and  the  center  of 
the  circle  inscribed  in  the  triangle. 

25.  Find  the  bisector  of  that  angle  between  the  lines  3  x  —  VSy+ 10 =0, 
V2  x  +  y  —  6  =  0in  which  the  origin  lies. 

26.  If  two  lines  are  given  in  the  normal  form,  what  is  represented  by 
their  sum  and  what  by  their  difference  ? 

27.  Show  that  the  angle  between  the  lines  x  +  y  =  0  and  x  —  y  =  0  is 
90°  whether  the  axes  are  rectangular  or  oblique. 


48  PLANE  ANALYTIC  GEOMETRY         [IIL  §  44 

44.  Pencils  of  Lines.  All  lines  through  one  and  the  same 
point  are  said  to  form  a  pencil;  the  point  is  called  the  center  of 
the  pencil.     If 

^^  \A'x-j-B'y-j-C  =  0 

are  any  two  different  lines  of  a  pencil,  the  equation 

(9)  Ax-{-By+  C-^k{A'x  +  B'y  +  C)  =  0, 

where  k  is  any  constant,  represents  a  line  of  the  pencil.  For, 
the  equation  (9)  is  of  the  first  degree  in  x  and  y,  and  the  coeffi- 
cients of  X  and  y  cannot  both  be  zero,  since  this  would  mean 
that  the  lines  (8)  are  parallel.  Moreover,  the  line  (9)  passes 
through  the  center  of  the  pencil  (8)  because  the  coordinates  of 
the  point  that  satisfies  each  of  the  equations  (8)  also  satisfy 
the  equation  (9). 

All  lines  parallel  to  the  same  direction  are  said  to  form  a 
pencil  of  parallels.  It  is  readily  seen  that  if  the  lines  (8)  are 
parallel,  the  equation  (9)  represents  a  line  parallel  to  them. 

EXERCISES 

1.  Find  the  line  :  (a)  through  the  point  of  intersection  of  the  lines 
4x— 7y+6  =  0,  6x-f-lly  —  7  =  0  and  the  origin  ;  (&)  through  the 
point  of  intersection  of  the  lines  4x  —  2y  —  3=0,  x  +  y  —  5  =  0  and 
the  point  (—2,  3);  (c)  through  the  point  of  intersection  of  the  lines 
4x— 6y4-6  =  0,  y— x  —  3  =  0,  of  slope  3 ;  (d)  through  the  intersection 
of5x— 6y-f-10  =  0,  2x  +  3y  —  12  =  0,  perpendicular  to  4  y  +  x  =  0. 

5.  Find  the  line  of  the  pencil  x  —  5  =  0,  y  +  2  =  0  that  is  inclined  to 
the  axis  Ox  at  30°. 

3.  Determine  the  constant  b  of  the  line  y  =  Sx  +  b  so  that  this  line 
shall  belong  to  the  pencil  3x  —  4y  +  6  =0,  x  =  5. 

4.  Find  the  line  joining  the  centers  of  the  pencils  x  —  3  y  =  12, 
5  X  -  2  y  =  1  and  x  +  y  =  6,  4x  —  5y  =  3. 

6.  Find  the  line  of  the  pencil  4x-5y-12  =  0,  3x  +  2y-16  =  0 
that  makes  equal  intercepts  on  the  axes. 


Ill,  §  45]  TWO  OR  MORE  LINES  49 

45.  Non-linear  Equations  representing  Lines.  When  two 
lines  are  given,  say 

A'x  +  B'y  +  O'  =  0, 
then  the  equation 

{Ac  +  By-^Cf){A'x  +  B'y-{-C)  =  0, 

obtained  by  multiplying  the  left-hand  members  (the  right-hand 
members  being  reduced  to  zero)  is  satisfied  by  all  the  points 
of  the  first  given  line  as  well  as  all  the  points  of  the  second 
given  line,  and  by  no  other  points. 

The  product  equation  which  is  of  the  second  degree  is  there- 
fore said  to  represent  the  two  given  lines.  Similarly,  by  equat- 
ing to  zero  the  product  of  the  left-hand  members  of  the  equations 
of  three  or  more  straight  lines  (whose  right-hand  members  are 
zero)  we  find  a  single  equation  representing  all  these  lines. 
An  equation  of  the  nth  degree  may  therefore  represent  n 
straight  lines,  viz.  when  its  left-hand  member  (the  right-hand 
member  being  zero)  can  be  resolved  into  n  linear  factors,  with 
real  coefiicients. 

EXERCISES 

1.  Find  the.  common  equation  of  the  two  axes  of  coordinates. 

2.  Show  that  n  lines  through  the  origin  are  represented  by  a  homo- 
geneous equation  (i.e.  one  in  which  all  terms  are  of  the  same  degree  in 
X  and  y)  of  the  nth  degree. 

3.  Draw  the  lines  represented  by  the  following  equations  : 
(or)  (X  -a)(y-b)=  0.  (/)  xy  -  ax  =  0. 

(&)  3  a;2  -  xy  -  4  2/2  =  0.  (g)  y^  -  ^y^  +  Qy  =  0. 

(c)  a;2_9y2-o.  (h)  x^y-xy  =  0. 

(d)  ax2  4-  &?/2  =  0.  (0   2/3  _  6  a:2/2  +  11  a;2y  -  6  x3  =  0. 

(e)  x2  -  X  -  12  =  0. 

4.  What  relation  must  hold  between  a,  h,  &,  if  the  lines  represented 
by  ax2  +  2  hxy  +  by^  =  0  are  to  be  real  and  distinct,  coincident,  imag- 
inary ? 


50  PLANE  ANALYTIC  GEOMETRY         [III,  §  45 

MISCELLANEOUS  EXERCISES 

1.  Find  the  angle  between  the  Hnes  represented  by  the  equation 
ax^  +  2  hxy  +  hy'^  =  0.  What  is  the  condition  for  these  Unes  to  be  per- 
pendicular ?  coincident  ? 

2.  Reduce  the  general  equation  ^x  +  J?«/  4-  C  =  0  to  the  normal 
form  x  cos  /3  +  2/  sin  /3  =  p  by  considering  that,  if  both  equations  represent 
the  same  line,  the  intercepts  must  be  the  same. 

3.  Find  the  line  through  (xi ,  y{)  making  equal  intercepts  on  the  axes. 

4.  Find  the  area  of  the  triangle  formed  by  the  Unes  y  —  m\x  +  6i , 
y  =  miX  -\-b2,y  =  b. 

6.    What  does  the  equation  <p  =  const,  represent  in  polar  coordinates  ? 

6.  Find  the  polar  equation  of  the  line  through  (6,  ir)  and  (4,  J  t). 

7.  Derive  the  determinant  expression  for  the  area  of  a  triangle  (§  14) 
by  multiplying  one  side  by  half  the  altitude. 

8.  The  weights  w?,  W  being  suspended  at  distances  d,  Z),  respectively, 
from  the  fulcrum  of  a  lever,  we  have  by  the  law  of  the  lever  WD  =  wd. 
If  the  weights  are  shifted  along  the  lever,  then  to  every  value  of  d  cor- 
responds a  definite  value  of  Z) ;  i.e.  Z>  is  a  function  of  d.  Represent  this 
function  graphically  ;  interpret  the  part  of  the  hne  in  the  third  quadrant. 

9.  A  train,  after  leaving  the  station  J.,  attains  in  the  first  6  minutes, 
IJ  miles  from  A,  the  speed  of  30  miles  per  hour  with  which  it  goes  on. 
How  far  from  A  will  it  be  50  minutes  after  starting?  (Compare  Ex- 
ample 4,  §  29.)    Illustrate  graphically,  taking  s  in  miles,  t  in  minutes. 

10.  A  train  leaves  Detroit  at  8  hr.  25  m.  a.m.  and  reaches  Chicago  at 
4  hr.  5  m.  p.m.  ;  another  train  leaves  Chicago  at  10  hr.  30  m.  a.m.  and 
arrives  in  Detroit  at  5  hr.  30  m.  p.m.  The  distance  is  284  miles.  Regard- 
ing the  motion  as  uniform  and  neglecting  the  stops,  find  graphically  and 
analytically  where  and  when  the  trains  meet.  If  the  scale  of  distances 
(in  miles)  be  taken  1/20  of  the  scale  of  times  (in  hours),  how  can  the 
velocities  be  found  from  the  slopes  ? 

11.  A  stone  is  dropped  from  a  balloon  ascending  vertically  at  the  rate 
of  24  ft. /sec;  express  the  velocity  as  a  function  of  the  time  (Example  6, 
§  29) .     What  is  the  velocity  after  4  sec.  ? 

12.  How  long  will  a  ball  rise  if  thrown  vertically  upward  with  an 
initial  velocity  of  100  ft. /sec.  ? 


CHAPTER   IV 
THE  CIRCLE 

46.   Circles.    A  circle,  in  a  given  plane,  is  defined  as  the  locus 
of  all  those  points  of  the  plane  which  are      y 
at  the  same  distance  from  a  fixed  point. 

Let  G  (h,  Tc)  be  the  center,  r  the  radius 
(Fig.  31) ;  the  necessary  and  sufficient 
condition  that  any  point  P  (x,  y)  is  at 
the  distance  r  from  C  (h,  k)  is  that  Fig.  31 


1-* 


(1)  (aJ  -  ^)2  +  (2/  -  A;)2  =  *'2. 

This  equation,  which  is  satisfied  by  the  coordinates  x,  y  of 
every  point  on  the  circle,  and  by  the  coordinates  of  no  other 
point,  is  called  the  equation  of  the  circle  of  center  C  (h,  k)  and 
radius  r. 

If  the  center  of  the  circle  is  at  the  origin  0  (0,  0),  the  equation 
of  the  circle  is  evidently 

(2)  a?-\-y'^  =  i\ 


EXERCISES 

Write  down  the  equations  of  the  following  circles : 

(a)   center  (3,  2),  radius  7  ; 

(6)   center  at  origin,  radius  3  ; 

(c)   center  at  (—  a,  0),  radius  a  ; 

{d)   circle  of  any  radius  touching  the  axis  Ox  at  the  origin  ; 

(e)   circle  of  any  radius  touching  the  axis  Oy  at  the  origin. 

Illustrate  each  case  by  a  sketch. 

51 


52  PLANE  ANALYTIC  GEOMETRY  [IV,  §  47 

47.  Equation  of  Second  Degree.  Expanding  the  equation 
(1)  of  §  46,  we  obtain  the  equation  of  the  circle  in  the  new  form 

x^^y'i-2hx-2  ky  +  li"  +  Ic"  -  r^  =  0. 
This  is  an  equation  of  the  second  degree  in  x  and  y.     But  it  is  of 
a  particular  form.     The  general  equation  of  the  second  degree 
in  X  and  y  is  of  the  form 

(3)  Ax'-{-2Hxy-\-By^-\-2Gx  +  2Fy-\-C=0'y 

i.e.  it  contains  a  constant  term,  (7;  two  terms  of  the  first  de- 
gree, one  in  x  and  one  in  y ;  and  three  terms  of  the  second  de- 
gree,  one  in  x^,  one  in  osy,  and  one  in  y\ 
If  in  this  general  equation  we  have 

it  reduces,  upon  division  by  Aj  to  the  form 

ii^  +  y^  +  --^x  +  —y-\--  =  0, 

which  agrees  with  the  form  (1)  of  the  equation  of  a  circle,  ex- 
cept for  the  notation  for  the  coefficients. 

We  can  therefore  say  that  any  equation  of  the  second  degree 
which  contains  no  xy-term  and  in  which  tlie  coefficients  of  a^  and 
y^  are  equal,  may  represent  a  circle. 

48.  Detennination  of  Center  and  Radius.  To  draw  the 
circle  represented  by  the  general  equation 

(4)  Ax^  +  Ay^  +  2  «a;  +  2  Fy  +  C  =  O, 

where  A,  G,  F,  Q  are  any  real  numbers  while  ^  ^  0,  we  first 
divide  by  ^  and  complete  the  squares  in  x  and  y ;  i.e.  we  first 
write  the  equation  in  the  form 

The  left-hand  member  represents  the  square  of  the  distance  of 
the  point  (x,  y)  from  the  point  (—  0/A,  —  F/A) ;  the  right- 


IV,  §49]  THE  CIRCLE  53 

hand  member  is  constant.    The  given  equation  therefore  repre- 
sents the  circle  whose  center  has  the  coordinates 


7.  ^    T.         ^ 


and  whose  radius  is 


This  radius  is,  however,  imaginary  if  G^  -^  F"^  <  AC;  in  this 
case  the  equation  is  not  satisfied  by  any  points  with  real  co- 
ordinates. 

If  G"^  +  F^^  AG)  the  radius  is  zero,  and  the  equation  is  satis- 
fied only  by  the  coordinates  of  the  point  ( —  G/A,  —  F/A). 

If  G^+F"^  >  AG,  the  radius  is  real,  and  the  equation  repre- 
sents a  real  circle. 

Thus,  the  general  equation  of  the  second  degree  (3),  §  47,  repre- 
sents d  circle  if,  and  only  if, 

A  =  B=^0,H=0,  G'  +  F'>AG. 

49.  Circle  determined  by  Three  Conditions.  The  equation 
(1)  of  the  circle  contains  three  constants  h,  7c,  r.  The  general 
equation  (4)  contains  four  constants  of  which,  however,  only 
three  are  essential  since  we  can  always  divide  through  by  one  of 
these  constants.  Thus,  dividing  by  A  and  putting  2  G/A  =  a, 
2  F/A  —  b,  C/A  =  c,  the  general  equation  (4)  assumes  the  form 
(5)  a^-f2/'  +  a^-h&2/  +  c  =  0, 

with  the  three  constants  a,  b,  c. 

The  existence  of  three  constants  in  the  equation  corresponds 
to  the  possibility  of  determining  a  circle  geometrically,  in  a 
variety  of  ways,  by  three  conditions.  It  should  be  remembered 
in  this  connection  that  the  equation  of  a  straight  line  contains 
two  essential  constants,  the  line  being  determined  by  two 
geometrical  conditions  (§  30). 


54  PLANE  ANALYTIC  GEOMETRY  [IV,  §  49 

EXERCISES 

1.  Draw  the  circles  represented  by  the  following  equations: 

(a)  2 a;2  +  2  2/2  _  8  X  +  5  y  +  1  =  0.     (b)  Sx^  +  Sy^  +  17  x  -  15y-6  =  0. 
(c)  4a;2  +  4y2_6x-10y +4  =  0.     (d)  x^ +  y^  +  x  -  iy  =  0. 
(e)  2 x2  +  2  y2 _  7  x  =  0.  (f)  x^  +  y'^-Sx-e=0. 

2.  What  is  the  equation  of  the  circle  of  center  (A,  k)  that  touches  the 
axis  Ox  ?  that  touches  the  axis  Oy  ?  that  passes  through  the  origin  ? 

3.  What  is  the  equation  of  any  circle  whose  center  lies  on  the  axis 
Ox  ?  on  the  axis  Oy  ?  on  the  line  y=  x?  on  the  line  y  =  2 x ?  on  the  line 
y  =  mx? 

4.  Find  the  equation  of  the  circle  whose  center  is  at  the  point  (—  4,  6) 
and  which  passes  through  the  point  (2,  0). 

5.  Find  the  circle  that  has  the  points  (4,  —  3)  and  (—  2,  —  1)  as  ends 
of  a  diameter. 

6.  A  swing  moving  in  the  vertical  plane  of  the  observer  is  48  ft.  away 
and  is  suspended  from  a  pole  27  ft.  high.  If  the  seat  when  at  rest  is  2  ft. 
above  the  ground,  what  is  the  equation  of  the  path  (for  the  observer  as 
origin)?  What  is  the  distance  of  the  seat  from  the  observer  when  the 
rope  is  inclined  at  45°  to  the  vertical  ? 

7.  Find  the  locus  of  a  point  whose  distance  from  the  point  (a,  6)  is  k 
times  its  distance  from  the  origin. 

Let  P  (x,  y)  be  any  point  of  the  locus  ;  then  the  condition  is 


V(x  -  a)2  +  (y  -  6)2  =  kVx^  +  y* . 

upon  squaring  and  rearranging  this  becomes : 

(1  -  k2)x2  +  (1  -  K'^)y^  -  2  ax  -  2  &y  +  a2  +  62  =  0. 

Hence  for  any  value  of  k  except  k  =  1,  the  locus  is  a  circle  whose  center  is 
a/(l  -  K^),  6/(1  —  /c2)  and  whose  radius  is  k  VaM^/(l  —  k^).  What 
is  the  locus  when  k  =  1  ? 

8.  Find  the  locus  of  a  point  twice  as  far  from  the  origin  as  from  the 
point  (6,  —  3).    Sketch. 

9.  What  is  the  locus  of  a  point  whose  distances  from  two  points  Pi, 
P2  are  in  the  constant  ratio  k  ? 


IV,  §  50]  THE  CIRCLE  55 

10.  Determine  the  locus  of  the  points  which  are  k  times  as  far  from 
the  point  (—2,  0)  as  from  the  point  (2,  0).  Assign  to  k  the  values 
V5,  V3,  V2,  I VS,  I V3,  I  \/2  and  illustrate  with  sketches  drawn  with 
respect  to  the  same  axes. 

11.  Determine  the  locus  of  a  point  whose  distance  from  the  line 
3x  —  4y+l=0  is  equal  to  the  square  of  its  distance  from  the  origin. 
Illustrate  with  a  sketch. 

12.  Determine  the  locus  of  a  point  if  the  square  of  its  distance  from 
the  line  x  +  y  —  a  =  0  is  equal  to  the  product  of  its  distances  from  the 
axes. 

50.  Circle  in  Polar  Coordinates.  Let  us  now  express  the 
equation  of  a  circle  in  polar  coordinates.  If  C(ri,  <^i)  is  the 
center  of  a  circle  of  radius  a  (Fig.  32) 
and  P(r,  <f>)  any  point  of  the  circle, 
then  by  the  cosine  law  of  trigo- 
nometry o^"'A^9i. 

r^  +  ri^  —  2  Tir  cos  (<^  —  <^i)  =  a\  Fig.  32 

This  is  the  equation  of  the  circle  since,  for  given  values  of  rj, 
<^i,  a,  it  is  satisfied  by  the  coordinates  r,  0  of  every  point  of 
the  circle,  and  by  the  coordinates  of  no  other  point. 
Two  special  cases  are  important : 

(1)  If  the  origin  0  be  taken  on  the  circumference  and  the 
polar  axis  along  a  diameter  OA  (Fig.  33), 
the  equation  becomes 

r"^  -\-  a^  —  2  ar  cos  <l>  =  a% 
i.e.  r  =  2a cos  <^. 

This  equation  has  a  simple  geometrical 

interpretation :  the  radius  vector  of  any  ^^' 

point  Pon  the  circle  is  the  projection  of  the  diameter  OA  =2  a 

on  the  direction  of  the  radius  vector. 

(2)  If  the  origin  be  taken  at  the  center  of  the  circle,  the 
equation  is  r  =  a. 


56  PLANE  ANALYTIC  GEOMETRY  [IV,  §  50 

EXERCISES 

1.  Draw  the  following  circles  in  polar  coordinates  : 

(a)  r  =  10 cos 0.  (6)  r  =  2a  cos  (0  -  iir).  (c)   r  =  sin0. 

id)  r  =  6.  (e)  r  =  7  sin  (0  _  ^  «■) .  (f)r  =  17  cos  0. 

2.  Write  the  equation  of  the  circle  in  polar  coordinates : 
(a)  with  center  at  (10,  ^  ir)  and  radius  5  ; 

(6)  with  center  at  (6,  \ir)  and  touching  the  polar  axis  ; 

(c)  with  center  at  (4,  f  w)  and  passing  through  the  origin  ; 

(d)  with  center  at  (3,  tt)  and  passing  through  the  point  (4,  |ir). 

3.  Change  the  equations  of  Ex.  (1)  and  (2)  to  rectangular  coordinates 
with  the  origin  at  the  pole  and  the  axis  Ox  coincident  with  the  polar  axis. 

4.  Determine  in  polar  coordinates  the  locus  of  the  midpoints  of  the 
chords  drawn  from  a  fixed  point  of  a  circle. 

51.  Intersection  of  Line  and  Circle.  To  solve  two  equa- 
tions in  X  and  y  of  which  one  is  of  the  first  degree  (linear) 
while  the  other  is  of  the  second  degree,  it  is  generally  most 
convenient  to  solve  the  linear  equation  for  either  x  or  y  and  to 
substitute  the  value  so  found  in  the  equation  of  the  second  degree. 
It  then  remains  to  solve  a  quadratic  equation. 

The  method  for  solving  a  quadratic  equation  consists  in 
completing  the  square  of  the  terms  in  x^  and  x.     The  equation 

aa;2  +  6a;  -f  c  =  0 
has  the  roots 


—  b±  V62  —  4  ac 

X  = • 

2a 

The  quantity  h^  —  4:ac  is  called  the  discriminant  of  the 
equation.  According  as  the  discriminant  is  positive,  zero,  or 
negative,  the  roots  are  real  and  different,  real  and  equal,  or 
imaginary. 

An  equation  of  the  first  degree  represents  a  straight  line. 
If  the  given  equation  of   the  second  degree  be  of   the  form 


IV,  §51]  THE  CIRCLE  57 

described  in  §  47,  it  will  represent  a  circle.  By  solving  two 
such  simultaneous  equations  we  find  the  coordinates  of  the 
points  that  lie  both  on  the  line  and  on  the  circle,  i.e.  the  points 
of  intersection  of  line  and  circle. 

Let  us  find  the  intersections  of  the  line 
y  =  mx  +  h 
with  the  circle  about  the  origin 

ic2  +  2/^  =  r"^' 
Substituting  the  value  of  y  from  the  former  equation  into  the 
latter,  we  find  the  quadratic  equation  in  x : 

fl;2  +  {mx  H-  by  =  r2, 
or  (1  +  m2)  a;2  +  2  mbx  +  &2  _  ,^2  ^  q. 

The  two  roots  iCi,  x^  of  this  equation  are  the  abscissas  of  the 
points  of  intersection ;  the  corresponding  ordinates  are  found 
by  substituting  x-^,  x^  in  ?/  =  mx  +  h. 

It  is  easily  seen  that  the  abscissas  x^,  X2  are  real  and  differ- 
ent if  (1  +  m2)  r2  -  62  >  0, 

...  b         ^ 

I.e.  II  — ==^  <  r. 

Vl  -1-  m2 


Since  m  =  tan  a,  and  hence  1/ Vl  +  m2  =  cos  a,  the  .preceding 
relation  means  that  b  cos  a  <r,  i.e.  the  line  has  a  distance  from 
the  origin  less  than  the  radius  of  the  circle.     If 
(l  +  m2)r2-52  =  0, 

the  roots  x^,  X2  are  real  and  equal.     The  line  and  the  circle 
then  have  only  a  single  point  in  common.     Such  a  line  is  said 
to  touch  the  circle  or  to  be  a  tangent  to  the  circle.     If 
(1  +  m2)  r2  _  62  <  0, 

the  roots  are  complex,  and  the  line  has  no  points  in  common 
with  the  circle. 


58  PLANE  ANALYTIC  GEOMETRY  [IV,  §  52 

52.  The  General  Case.  The  intersections  of  the  line  and 
circle 

are  found  in  the  same  way :  substitute  the  value  of  y  (or  x), 
found  from  the  equation  of  the  line,  in  the  equation  of  the 
circle  and  solve  the  resulting  quadratic  equation. 

It  is  often  desired  to  determine  merely  vjJiether  the  line  is 
tangent  to  the  cirde.  To  answer  this  question,  substitute  y 
(or  x)  from  the  linear  equation  in  the  equation  of  the  circle 
and,  without  solving  the  quadratic  equation,  write  down  the  con- 
dition for  equal  roots  (b^  =  4  ac,  §  51). 

EXERCISES 

1.  Find  the  coordinates  of  the  points  where  the  circle  x^  -{■  y^  —  x  +  y 
—  12  =  0  crosses  the  axes. 

2.  Find  the  intersections  of  the  line  3x  +  y— 5  =  0  and  the  circle 
a;2  -f  y2  _  22  X  -  4  y  +  25  =  0. 

3.  Find  the  intersections  of  the  line  2x  —  7y  +  5  =  0  and  the  circle 
2 x2  +  2  y2  +  9x  +  9  y  -  11  =  0. 

4.  Find  the  equations  of  the  tangents  to  the  circle  x^  +  y'^  =  16  that 
are  parallel  to  the  line  y  =  —  3  x  +  8. 

6.  Show  that  the  equations  of  the  tangents  to  the  circle  x^  -{■  y^  =  r^ 
with  slope  n»  are  y  =  mx  ±  rVl  +  m^. 

6.  For  what  value  of  r  will  the  line  3  x-  2  y  —  5  =  0  be  tangent  to  the 
circle  x'^  +  y^  =  r^? 

7.  Find  the  equations  of  the  tangents  to  the  circle  2x^-\-2y^  —  3x 
+  6y  —  7  =  0  that  are  perpendicular  to  the  line  x+2y  +  3  =  0. 

8.  Find  the  midpoint  of  the  chord  intercepted  by  the  line  6x-y  +  9=0 
on  the  circle  x^  +  y^  =  18. 

9.  Find  the  equations  of  the  tangents  to  the  circle  x^-\-y'^  =  58  that 
pass  through  the  point  (10,  4). 


IV,  §  54] 


THE  CIRCLE 


59 


63.  The  Tangent  to  a  Circle.  The  tangent  to  a  circle  (com- 
pare §  40)  at  any  point  P  may  be  defined  as  the  perpendicular 
through  P  to  the  radius  passing  through  P.  To  find  the  equa- 
tion of  the  tangent  to  a  circle  whose  center  is  at  the  origin, 

a;2  -f  2/2  =  r^, 
at  the  point  P  (x,  y)  of  the  circle  (Fig.  34),  observe  that  the 
distance  p  of  the  tangent  from  the  origin 
is  equal  to  the "  radius  r  and  that  the 
angle  p  made  by  this  distance  with  the 
axis  Ox  is  such  that 

cos  «  =  - ,  sm  «  =  ^ : 
r  r 

substituting  these  values  in  the  normal 

form  X  cos  ^  -f  Y"  sin  ^  =  p   of   the 

equation  of  a  line  (§  40),  we  find  as  equation  of  the  tangent 

xX^yY^T^, 

where  a;,  y  are  the  coordinates  of  the  point  of  contact  P  and 
X,  Y  are  those  of  any  point  of  the  tangent. 

64.  The  General  Case.  To  find  the  equation  of  the  tangent 
to  a  circle  whose  center  is  not  at  the  origin  let  us  write  the 
general  equation  (4),  §  48,  viz. 

(4)  Ax^^Ay''^2Qx-\-2Fy^C^% 

in  the  form 


Fig.  34 


{'-ih{' 


where  —  G/A,  —  F/A  are  the  coordinates  of  the  center  and 
G^/A"  +  F^/A^-  C/A  is  the  square  of  the  radius  r  (§  48). 
With  respect  to  parallel  axes  through  the  center  the  same  circle 
has  the  equation 


60  PLANE  ANALYTIC  GEOMETRY  [IV,  §  54 

and  the  tangent  at  the  point  P(x,  y)  of  the  circle  is  (§  63) : 

Hence,  transferring  back  to  the  original  axes,  we  find  as 
equation  of  the  tangent  at  P(x,  y)  to  the  circle  (4) : 

AxX  +  AyT+  G  (x-^X)-\-  F(y  +Y)+C=0. 

This  general  form  of  the  tangent  is  readily  remembered  if  we 
observe  that  it  can  be  derived  from  the  equation  (4)  of  the 
circle  by  replacing  x^  by  xXj  y^  by  yY,  2xby  x+X,  2yhyy-^T. 

EXERCISES 
1.  Find  the  tangent  to  the  given  circle  at  the  given  point : 
(a)  a;2  +  y2  =  4i,  (5,  _4). 
^b)  x^  +  y^  +  Gx  +  5y-lQ  =  0,  (-2,3). 

(c)  3x2 +  3y2  +  iOa;  +  17y  + 18  =  0,  (-2,  -5). 

(d)  x^  +  y^-ax-by=  0,  (a,  6). 

5.  The  equation  of  any  circle  through  the  origin  can  be  written  in  the 
form  (§  49)  x^ -{■  y^  +  ax  +  by  =  0  ;  show  that  the  line  ax  -\- by  =  0  is  the 
tangent  at  the  origin,  and  find  the  equation  of  the  parallel  tangent. 

3.  Derive  the  equation  of  the  tangent  to  the  circle  (x—h)^+{y—k)^  =  i^. 

4.  Show  that  the  circles  x^  +  y^  -  Qx +  2y +  2  =  0  and  x^  +  ys  _  4  y 
+  2  =  0  touch  at  the  point  (1,  1). 

6.  Find  the  tangents  to  the  circle  x^-\-y^  —  2x-10y-\-d  =  0  at  the 
extremities  of  the  diameter  through  the  point  (—  1,  11/2). 

6.  The  line  2  a;  +  y  =  10  is  tangent  to  the  circle  x^  +  y^  =  20;  what  is 
the  point  of  contact  ? 

7.  What  is  the  point  of  contact  if  ^4- -By +0  =  0  is  tangent  to  the 
circle  x^-\-y^  =  r^? 

8.  Show  that  x  —  y—l  =  0  is  tangent  to  the  circle  x^  -\^y^  +  ix 
—  10  y  —  3  =  0,  and  find  the  point  of  contact. 

9.  By  §  51,  the  line  y  =  mx  -\-  b  has  but  one  point  in  common  with 
the  circle  x^  -{^y^  =  r^  il  (I  +  m^)r^  =  b^  ;  show  that  in  this  case  the  radius 
drawn  to  the  common  point  is  perpendicular  to  the  line  y  =  mx  +  b. 


IV,  §  55] 


THE  CIRCLE 


61 


55.  Circle  through  Three  Pomts.  To  find  the  equation  of 
the  circle  passing  through  three  points  P^  {x^ ,  2/1),  P2  (^2  >  2/2)* 
P3  (x^ ,  2/3),  observe  that  the  coordinates  of  these  points  satisfy 
the  equation  of  the  circle  (§  49)  * 

(6)  x'  +  if  +  ax  +  hy  +  c^O) 

hence  we  must  have 


(J) 


^i  +  Vi  +  «^i  +  &2/1  +  c  =  0, 
^2  +  2/2^  +  ax^  +  &?/2  4-c  =  0, 

a^3^  +  2/3^  +  «i«3  +  ^2/3+  c  =  0. 


From  the  last  three  equations  we  can  find  the  values  of  a,  &, 
and  c ;  these  values  must  then  be  substituted  in  the  first  equar 
tion. 

In  general  this  is  a  long  and  tedious  operation.  What  we 
actually  wish  to  do  is  to  eliminate  a,  h,  c  between  the  four 
equations  above.  The  theory  of  determinants  furnishes  a  very 
simple  means  of  eliminating  four  quantities  between  four 
homogeneous  linear  equations.  Our  equations  are  not  homo- 
geneous in  a,  6,  c.  But  if  we  write  the  first  two  terms  in 
each  equation  with  the  factor  1 :  {x^  -f  y^)  •  1,  (x^  -f  y^)  •  1,  etc., 
we  have  four  equations  which  are  linear  and  homogeneous  in  1, 
a,  b,  c;  hence  the  result  of  eliminating  these  four  quantities  is 
the  determinant  of  their  coefficients  equated  to  zero.  Thus  the 
equation  of  the  circle  through  three  points  is 


a^  -j-  2/'      X     y 

1 

^i  +  2/1'     «i     2/1 

1 

^i  +  2/2'     ^2    2/2 

1 

^z+Vz      ^z     2/3 

1 

=  0. 


Compare  §  34,  where  the  equation  of  the  straight  line  through 
two  points  is  given  in  determinant  form. 


62  PLANE  ANALYTIC  GEOMETRY  [IV,  §  55 

EXERCISES 

1.  Find  the  equations  of  the  circles  that  pass  through  the  points : 
(a)  (2,3),  (-1,2),  (0,-3). 

(6)  (0,0),  (1,-4),  (5,0). 
(c)  (0,  0),  (a,  0),  (0,  6). 

2.  Find  the  circles  through  the  points  (3,-1),  (—1,-2)  which 
touch  the  axis  Ox. 

3.  Find  the  circle  through  the  points  (2,  1),  (-  1,  3)  with  center  on 
the  line  3x-y  +  2=0. 

4.  Find  the  circle  whose  center  is  (3,  —  2)  and  which  touches  the 
line  3a;  +  4y-12  =  0. 

6.   Find  the  circle  through  the  origin  that  touches  the  line 
4a; -5y- 14  =  Oat  (0,  2). 

6.  Find  the  circle  inscribed  in  the  triangle  determined  by  the  lines 

24x-72/+3=0,  3x-4y-9  =  0,   5a;  +  12y-50  =  0. 

7.  Two  circles  are  said  to  be  orthogonal  if  their  tangents  at  a  point  of 
intersection  are  perpendicular ;  the  square  of  the  distance  between  their 
centers  is  then  equal  to  the  sum  of  the  squares  of  their  radii.  If  the 
equations  of  two  intersecting  circles  are 

x^  -i-y^  +  aix  +  biy  +  Ci  =  0,  and  x^  +  y^  +  a^x  +  62^  +  c^  =  0, 
show  that  the  circles  are  orthogonal  when  aiOa  +  &162  =  2(Ci  -f  Ca). 

8.  Find  the  circle  that  has  its  center  at  (—2,  1)  and  is  orthogonal  to 
the  circle  x^  +  y  «_  q  a;  +  3  =  0. 

9.  Find  the  circle  that  has  its  center  on  the  line  y  =  3  a;  +  4,  passes 
through  the  point  (4,  —  3),  and  is  orthogonal  to  the  circle 

a;2  +  y^  +  13  a;  +  5  y  +  2  =0. 

66.    Inversion.      A  circle  of  center  O  and  radius  a  being  given 
(Fig.  35),  we  can  find  to  every  point  P  of  the  plane 
(excepting  the  center  0)  one  and  only  one  point  P 
on  OP^  produced  beyond  P  if  necessary,  such  that     /  y/^f 

OP .  OP'  =  a\ 

The  point  P'  is  said  to  be  inverse  to  P  with  respect  

to  the  circle  (0,  a) ;   and  as  the  relation  is  not  Fig.  35 


IV,  §57]  THE  CIRCLE  63 

changed  by  interchanging  P  and  P',  the  point  P  is  inverse  to  P^  The 
point  0  is  called  the  center  of  inversion. 

It  is  clear  that  (1)  the  inverse  of  a  point  P  within  the  circle  is  a  point 
P'  without,  and  vice  versa  ;  (2)  the  inverse  of  a  point  of  the  circle  itself 
coincides  with  it ;  (3)  as  P  approaches  the  center  0,  its  inverse  P'  moves 
off  to  infinity,  and  vice  versa. 

The  inverse  of  any  geometrical  figure  (line,  curve,  area,  etc.)  is  the 
figure  formed  by  the  points  inverse  to  all  the  points  of  the  given  figure. 

57.  Inverse  of  a  Circle.  Taking  rectangular  axes  through  O 
(Fig.  36),  we  find  for  the  relations  between  the  coordinates  of  two  in- 
verse points  P{x,  y),  P'  (x",  y')^  if  we  put  OP  =  r,  OP'  =  r'  i 

x'  _y'  _r'  _  rr'  _  a^ 
X     y      r      f^      r^ 
since  rt'  =:a^;  hence  / 


.     a^^       yf^     a^y 

a;2  +  y2'  ^      x^  +  y^ 


and  similarly 
x  = 


These  equations  enable  us  to  find  to  any  curve  whose  equation  is  given  the 
equation  of  the  inverse  curve,  by  simply  substituting  for  x,  y  their  values. 

Thus  it  can  be  shown  that  by  inversion  any  circle  is  transformed  into 
a  circle  or  a  straight  line. 

For,  if  in  the  general  equation  of  the  circle 

A(x^  -{-y'^)+2Gx-^2Fy-\-  C  =  0 
we  substitute  for  x  and  y  the  above  values,  we  find 

{x'^  +  2/'2)2  ^  ic'2  +  ?/'2  ^  x'2  +  y'^  ' 

that  is,  Aa^  +  2  Ga'^x'  +  2  Fa^  +  C{x'-^  +  y'^)  =  0, 

which  is  again  the  equation  of  a  circle,  provided  G  =^0.  In  the  special 
case  when  C  =  0,  the  given  circle  passes  through  the  origin,  and  its  in- 
verse is  a  straight  line.  Thus  every  circle  through  the  origin  is  trans- 
formed by  inversion  into  a  straight  line.  It  is  readily  proved  conversely 
that  every  straight  line  is  transformed  into  a  circle  passing  through  the 
origin  ;  and  in  particular  that  every  line  through  the  origin  is  transformed 
into  itself,  as  is  obvious  otherwise. 


64 


PLANE  ANALYTIC  GEOMETRY  [IV,  §  57 


EXERCISES 

1.  Find  the  coordinates  of  the  points  inverse  to  (4,  3),  (2,  0),  (—  5,  1) 
with  respect  to  the  circle  x^  +  i/2  =  25. 

2.  Show  that  by  inversion  every  hue  (except  a  line  through  the  center) 
is  transformed  into  a  circle  passing  through  the  center  of  inversion. 

3.  Show  that  all  circles  with  center  at  the  center  of  inversion  are 
transformed  by  inversion  into  concentric  circles. 

4.  Find  the  equation  of  the  circle  about  the  center  of  inversion  which 
is  transformed  into  itself. 

6.   With  respect  to  the  circle  x^-\-y^  =  16,  find  the  equations  of  the 
curves  inverse  to  : 

(a)  x=5,      {h)  x-y=0,      (c)  x^-\-y^-6x=0,      (d)  x'^-\-y^-10y-{-l=0, 
(e)  3x-4y+6=0. 

6.  Show  that  the  circle  Ax^ -\-  Ay^ +  2  Gz +  2  Fy  +  a^A  =  0  is  trans- 
formed into  itself  by  inversion  with  respect  to  the  circle  x^ -\-  y"^  =  a*. 

7.  Prove  the  statements  at  the  end  of  §  57. 

58.    Pole  and  Polar.     Let  P,  P'  (Fig.  37)  be  inverse  points  with 
respect  to  the  circle  (O,  a)  ;  then  the  perpen- 
dicular I  to  OP  through  P'  is  called  the  polar  of 
P,  and  P  the  pole  of  the  line  Z,  with  respect  to 
the  circle. 

Notice  that  (1)  if  (as  in  Fig.  37)  Plies  within 
the  circle,  its  polar  I  lies  outside ;  (2)  if  P  lies 
outside  the  circle,  its  polar  intersects  the  circle 
in  two  points ;  (3)  if  P  lies  on  the  circle,  its 
polar  is  the  tangent  to  the  circle  at  P. 

Referring  the  circle  to  rectangular  axes  through  its  center  (Fig.  38)  so 
that  its  equation  is 

x2  +  2/2  =  a2, 

we  can  find  the  equation  of  the  polar  I  of 
any  given  point  P  (x,  y).  For,  using 
as  equation  of  the  polar  the  normal 
form  X  cos  /3  +  F  sin  /3  =  p,  we  have 
evidently,  if  P'  is  the  point  inverse 
toP: 


Fig.  37 


r 

/»^>i\ 

^ 

I 

^y\   l\ 

JC 

M 

Fig.  38 

\ 

IV,  §  59] 

cos/3  = 


THE  CIRCLE 


65 


sin/3 


y 


therefore  the  equation  becomes 

xX      .       yY      _ 


,  p=  0P'  = 


■N/aj2  +  2/2 


or  simply 


Va;2  +  2/2      Va;2  +  2/2      Vx^  +  2/2 
xX+2/r=a2. 


This  then  is  the  equation  of  the  polar  I  of  the  point  P  (x,  y)  with  re- 
spect to  the  circle  of  radius  a  about  the  origin.  If,  in  particular,  the 
point  P  (x,  y)  lies  on  the  circle,  the  same  equation  represents  the  tan- 
gent to  the  circle  x^  +  2/^  =  a^  at  the  point  P  (x,  y),  as  shown  previously 
in  §  53. 

59.  Chord  of  Contact.  The  polar  l  of  any  outside  point  P  with 
respect  to  a  given  circle  passes  through  the  points  of  contact  Ci ,  C2  of 
the  tangents  drawn  from  P  to  the  circle. 

To  prove  this  we  have  only  to  show  that  if  Ci  is  one  of  the  points  of 
intersection  of  the  polar  I  of  P  with  the  circle,  then  the  angle  OCiP 
(Fig.  39)  is  a  right  angle.     Now  the  triangles 
OCiP  and  OP'C\  are  similar  since  they  have 
the  angle  at  0  in  common  and  the  including 
sides  proportional  owing  to  the  relation 

OP-  0P'  =  a\ 

OP  ^    a 
a       OP'' 


i.e 


It  follows  that  '^  OCiP  = 


Fig.  39 


where  a  =  OCi. 
OP'Gi  =  ^w. 

The  rectilinear  segment  C1O2  is  sometimes  called  the  chord  of  contact 
of  the  point  P.  We  have  therefore  proved  that  the  chord  of  contact  of 
any  outside  point  P  lies  on  the  polar  of  P. 

It  follows  that  the  equations  of  the  tangents  that  can  he  drawn  from 
any  outside  point  P  to  a  given  circle  can  be  found  by  determining  the 
intersections  Ci ,  d  of  the  polar  of  P  with  the  circle ;  the  tangents  are 
then  obtained  as  the  lines  joining  C\ ,  C2  to  P. 


66 


PLANE  ANALYTIC  GEOMETRY  [IV,  §  60 


60.  The  General  Case.  The  equation  of  the  polar  of  a  point 
P  (x,  y)  with  respect  to  any  circle  given  in  the  general  form  (4), 
§  48,  viz., 

(4)  Ax'^  +  Ay^-\-2Gx-Sr2Fy  +  C  =  ^, 

is  found  by  the  same  method  that  was  used  in  §  54  to  generalize  the 
equation  of  the  tangent.  Thus,  with  respect  to  parallel  axes  through  the 
center  the  equation  of  the  circle  is 

^^~A^^A^     A' 
the  equation  of  the  polar  of  P  (x,  y)  with  respect  to  these  axes  is  by 


58 


xX+yY: 


A^     A^ 


Hence,  transferring  back  to  the  original  axes,  we  find  as  equation  of  the 
polar  of  P  (a;,  y)  with  respect  to  the  circle  (4) : 

AxX+AyT+Gix  +  X)+F{y+  r)+C  =  0. 
If,  in  particular,  the  point  P  (x,  y)  lies  outside  the  circle,  this  polar 
contains  the  chord  of  contact  of  P ;  if  P  lies  on  the  circle,  the  polar  be- 
comes the  tangent  at  P  (§  54). 

61.  Construction  of  PolarS.  if  a  point  Pi  describes  a  line  I,  its 
polar  h  with  respect  to  a  given  circle  (0,  a)  turns  about  a  fixed  point, 
viz. ,  the  pole  P  of  the  line  I  (Fig.  40) . 
Conversely,  if  a  line  h  turns  about  one 
of  its  points  P,  its  pole  Pi  with  respect 
to  a  given  circle  (O,  a)  describes  a  line  I, 
viz.  the  polar  of  the  point  P. 

For,  the  line  I  is  transformed  by  in- 
version with  respect  to  the  circle  (0,  a) 
into  a  circle  passing  through  O  and 
through  the  pole  P  of  I;  as  this  circle 
must  obviously  be  symmetric  with  respect 
to  OP  it  must  have  OP  as  diameter.  Any 
point  Pi  of  I  is  transformed  by  inversion 
into  that  point  Q  of  the  circle  of  diameter  OP  at  which  this  circle  is  in- 
tersected by  OPi  .  The  polar  of  Pi  is  the  perpendicular  through  Q  to 
OPi ;  it  passes  therefore  through  P,  wherever  Pi  be  taken  on  I. 

The  proof  of  the  converse  theorem  is  similar. 


Fia.  40 


IV,  §61]  THE  CIRCLE  67 

The  pole  Pi  of  any  line  h  can  therefore  be  constructed  as  the  intersec- 
tion of  the  polars  of  any  two  points  of  h  ;  this  is  of  advantage  when  the 
line  h  does  not  meet  the  circle.  And  the  polar  li  of  any  point  Pi  can  be 
constructed  as  the  line  joining  the  poles  of  any  two  lines  through  Pi ;  this 
is  of  advantage  when  the  point  Pi  lies  inside  the  circle. 

EXERCISES 

1.  Find  the  equation  of  the  polar  of  the  given  point  with  respect  to 
the  given  circle  and  sketch  if  possible ; 

(a)   (4,  7),  0^2  + y2^  8. 

(6)   (0,  0),  x2  +  2/2  -  3  a;  -  4  =  0. 

(c)  (2,l),x^  +  y^-ix-2y+l=0. 

(d)  (2,  -  3),  x2  +  ?/2  4-  3  a;  +  10  2/  +  2  =  0. 

2.  Find  the  pole  of  the  given  line  with  respect  to  the  given  circle  and 
sketch  if  possible : 

(a)  X  +  2  y-  20  =  0,  x^  +  y"^  =  20. 

(b)  x  +  y  +  l=0,  x^  +  y^  =  4. 

(c)  ^x-y  =  19,  a;2  +  y2  =  25. 

(d)  Ax  +  By  +  C  =  0,  x^  +  y^  =  r^. 

(e)  y  =  mx  -\-  b,  x^  -\-  y^  =  r^. 

3.  Find  the  pole  of  the  line  joining  the  points  (20,  0)  and  (0,  10), 
with  respect  to  the  circle  x^  +  y^  =  25. 

4.  Find  the  tangent  to  the  circle  x2+?/2_io  a:+4  ?/+9=0  at  (7,  -  6). 
6.  Find  the  intersection  of  the  tangents  to  the  circle  2  a;2  +  2  ?/2—  15  cc 

+  2/  —  28  =  0  at  the  points  (3,  5)  and  (0,  —  4) . 

6.  Find  the  tangents  to  the  circle  x^-{-y^  — 6x  —  10y  +  2  =  0  that 
pass  through  the  point  (3,  —  3) . 

7.  Find  the  tangents  to  the  circle a;2  +  ?/2_3ic  +  ?/—  10  =  0  that  pass 
through  the  point  ( —  |,  —  V)  • 

8.  Show  that  the  distances  of  two  points  from  the  center  of  a  circle 
are  proportional  to  the  distances  of  each  from  the  polar  of  the  other. 

9.  Show  analytically  that  if  two  points  are  given  such  that  the  polar 
of  one  point  passes  through  the  second  point,  then  the  polar  of  the  second 
point  passes  through  the  first  point. 

10.  Find  the  poles  of  the  lines  x  —  y  —  S  =  0  and  x  +  y  +  8  =  0  with 
respect  to  the  circle  x'^  +  y'^  —  Qx  +  iy  +  S=0. 


68 


PLANE  ANALYTIC  GEOMETRY 


[IV,  §  62 


62.  Power  of  a  Point,     if  in  the  left-hand  member  of  the  equa- 
tion of  the  circle 

we  substitute  for  x  and  y  the  coordinates  Xi ,  j/i  of  a  point  Pi  not  on  the 
circle  (Fig.  41),  the  expression  {xi  —  h)^  +  {yi  —  ky  —  r^  is  different 
from  zero.  Its  value  is  called  the  power 
of  the  point  P\  (xi ,  y{)  with  respect  to 
the  circle.  As  {x^  —  hy  +  (yi  -  ky  is 
the  square' of  the  distance  PiC  =  d  be- 
tween the  point  Pi  (a;i,  yi)  and  the 
center  C{h,  k),  the  power  of  the  point 
-Pi  (a^ii  yO  with  respect  to  the  circle  is 
cP  —  r^ ;  and  this  is  positive  for  points 
without  the  circle  (d  >  r) ,  zero  for  points  Fig.  41 

on  the  circle  (d  =  r),  and  negative  for  points  within  the  circle  (d  <  r). 
If  the  point  lies  without  the  circle,  its  power  has  a  simple  interpretation  ; 
it  is  the  square  of  the  segment  PiT  =  t  oi  the  tangent  drawn  from  Pi  to 

the  circle : 

t'i=cP-r^=(^Xi-  hy  -\-  (yi  -  ky  -  r^. 

Hence  the  length  t  of  the  tangent  that  can  be  drawn  from  an  outside 
point  Pi  {xi ,  j/i)  to  a  circle  x^  +  y"^  +  ax  -\- by  -^  c  =  0  is  given  by 

«2  =  xi^  +  yi*  -f-  aa;i  -f  &yi  +  c. 

Notice  that  the  coeflBcients  of  x^  and  y^  must  be  1.    Compare  the  similar 
case  of  the  distance  of  a  point  from  a  line  (§  42). 

63.  Radical  Axis.     The  locus  of  a  point  whose  powers  with  respect 

to  any  two  circles 

a;2  +  r/2  +  aix  -\-  Iny  -I-  Ci  =  0, 

x2  -}-  y2  -f  aax  +  b-2y  -|-  C2  =  0, 
are  equal  is  given  by  the  equation 

a;2  +  y2  ^  aix  +  biy  -{- ci  =  x^ -\- y^  +  a^x  -H  b^y  +  c^., 
which  reduces  to 

(ai  -  a2)x  -}-  (61  -  b2)y  +  (ci  -  C2)  =  0. 
This  locus  is  therefore  a  straight  line  ;  it  is  called  the  radical  axis  of  the 
two  circles.     It  always  exists  unless  ai  =  a-z  and  bi  =  bo,  i.e.  unless  the 
circles  are  concentric. 


IV,  §64]  THE  CIRCLE  69 

Three  circles  taken  in  pairs  have  three  radical  axes  which  pass  through 
a  common  point,  called  the  radical  center.  For,  if  the  equation  of  the 
third  circle  is  x^  +  y^  +  a,x  +  h,y  +  C3  =  0, 

the  equations  of  the  radical  axes  will  be 

(.052  -  «3)a;  +  (62  -  63)2/ +  (C2  -  C3)  =  0, 
{az  -  a{)x  +  (63  -  hi)y  +(03  -  Ci)  =  0, 
(«!  -  a2)a;  +  (&i  — &2)y +(ci  — C2)  =  0. 
These  lines  intersect  in  a  point,  since  the  detenninant  of  the  coefficients 
in  these  equations  is  equal  to  zero  (Ex.  3,  p.  38). 

64.  Family  of  Circles.    The  equation 

(8)  (x^  +  2/2  +  aix  +  hiy  +  ci)  +  k{x^  +  y^  +  a2X  +  h^y  +  C2)  =  0 
represents  a  family^  or  pencil^  of  circles  each  of  which  passes  through  the 
points  of  intersection  of  the  circles 

(9)  x^  +  y^  +  aix  +  biy  +  Ci  =  0, 
and 

(10)  x'^-\-y^+  a^x  +  h^y  +  Ca  =  0, 

if  these  circles  intersect.     For,  the  equation  (8)  written  in  the  form 

(1  +  /c)x2  +(1  +  /c)y2  +(ai  +  Ka'i)x  +(&i  +  Khi)y  +  Ci  +  kC2  =  0 
represents  a  circle  for  every  value  of  k  except  /c=  —  1,  as  the  coefficients 
of  x2  and  y"^  are  equal  and  there  is  no  x?/-term  (§47).  Each  one  of  the 
circles  (8)  passes  through  the  common  points  of  the  circles  (9)  and  (10) 
if  they  have  any,  since  the  equation  (8)  is  satisfied  by  the  coordinates 
of  those  points  which  satisfy  both  (9)  and  (10).  Compare  §  44.  The 
constant  k  is  called  the  parameter  of  the  family. 

In  the  special  case  when  k  =—  1,  the  equation  is  of  the  first  degree 
and  hence  represents  a  line,  viz.  the  radical  axis  (§  63)  of  the  two  circles 
(9),  (10).  If  the  circles  intersect,  the  radical  axis  contains  their  com- 
mon chord. 

EXERCISES 

1.  Find  the  powers  of  the  following  points  with  respect  to  the  circle 
x2  +  y2  _  3  3.  _  2  y  =  0  and  thus  determine  their  positions  relative  to  the 
circle:    (2,0),  (0,0),  (0,  -4),  (3,2). 

2.  What  is  the  length  of  the  tangent  to  the  circle :  (a)  x"^  -\-  y^  -\-  ax 
+  62/  +  c  =  0  from  the  point  (0,  0),  (6)  (x  -  2)2+  (^  -  3)2  -  1  =  0  from 
the  point  (4,  4)  ? 


70  PLANE  ANALYTIC  GEOMETRY  [IV,  §  64 

3.  By  §  62,  t^  =  d:^  —  r'^=(d-\-  r){d  —  r);  interpret  this  relation 
geometrically. 

4.  Find  the  radical  axis  of  the  circles  x^  '\-  y^  -\-  ax -\-  by  -\-  c  =  0  and 
x^  +  y^  +  bx  -\-  ay  +  c  =  0  and  the  length  of  the  common  chord. 

6.  Find  the  radical  center  of  the  circles  x^  +  y2  ^  Sx-\-4y  —  'J=0, 
x^  -{■y^  =  l6,  2{x^  +  y^)  -{■  Q  X  -\-  1  =  0.  Sketch  the  circles  and  their  radi- 
cal axes. 

6.  Find  the  circle  that  passes  through  the  intersections  of  the  circles 
x^  -\-y'^+  5x  =  0  and  x^  +y^-{-x  —  2y-6  =  0,  and  (a)  passes  through 
the  point  (—  5,  6),  (6)  has  its  center  on  the  line  4x  —  2y  —  16  =  0, 
(c)  has  the  radius  6. 

7.  Sketch  the  family  of  circles  x^  +  y^  -  6  y  +  k(x^ -\-y'^ -\- By)  =  0. 

8.  What  family  of  circles  does  the  equation  Az  -\-  By  -\-  C  +  k{x^ 
4-  y2  4-  (jx  +  &?/  +  c)  =  0  represent  ? 

9.  Find  the  family  of  curves  inverse  to  the  family  of  lines  y  =  mx  +  b\ 
(a)  with  m  constant  and  b  variable,  (6)  with  m  variable  and  b  constant. 
Draw  sketches  for  each  case. 

10.  Show  that  a  circle  can  be  drawn  orthogonal  to  three  circles,  pro- 
vided their  centers  are  not  in  a  straight  line. 

11.  Find  the  locus  of  a  point  whose  power  with  respect  to  the  circle 
2x2  +  2y2_5a;  +  lly  —  6=0is  equal  to  the  square  of  its  distance  from 
the  origin.     Sketch. 

12.  Find  the  locus  of  a  point  if  the  sum  of  the  squares  of  its  distances 
from  the  sides  of  an  equilateral  triangle  of  side  2  a  is  constant. 

15.  Show  that  the  circle  through  the  points  (2,  4),  (-  1,  2),  (3,  0)  is 
orthogonal  to  the  circle  which  is  the  locus  of  a  point  the  ratio  of  whose 
distances  from  the  points  (2,  3)  and  (—  1,  2)  is  3.     Sketch. 

14.  Show  that  the  circles  through  two  fixed  points,  say  (—a,  0), 
(a,  0),  form  a  family  like  that  of  Ex.  8. 

16.  The  locus  of  a  point  whose  distances  from  the  fixed  points  (—a,  0), 
(a,  0)  are  in  the  constant  ratio  *c  ( ^  1)  is  the  circle 

x^  +  y'^  +  2^-^ax  +  a^  =  0. 
1  —  k'^ 

Compare  Ex.  9,  p.  54.  Show  that,  whatever  k{=^1),  this  circle  inter- 
sects every  circle  of  the  family  of  Ex.  15  at  right  angles. 


CHAPTER   V 

POLYNOMIALS 

PART  I.     QUADRATIC   FUKCTION  —  PARABOLA 

65.  Linear  Function.  As  mentioned  in  §  28,  an  expression 
of  the  form  mx  -\-  h,  where  m  and  6  are  given  real  numbers 
(m^O)  while  a;  may  take  any  real  value,  is  called  a  linear 
function  of  x.  We  have  seen  that  this  function  is  represented 
graphically  by  the  ordinates  of  the  straight  line 

y  =  mx  -\-  b ; 

b  is  the  value  of  y  for  x  =  0,  and  m  is  the  slope  of  the  line,  i.e. 
the  rate  of  change  of  the  function  y  with  respect  to  x. 

66.  Quadratic  Function.  Parabola.  An  expression  of 
the  form  ax"^  +  &a;  +  c  in  which  a  :^  0  is  called  a  quadratic  func- 
tion of  Xj  and  the  curve 

y  —  ax"^ -{- bx -\-  c, 

whose  ordinates  represent  the  function,  is  called  a  parabola. 

If  the  coefficients  a,  6,  c  are  given  numerically,  any  number 
of  points  of  this  curve  can  be  located  by  arbitrarily  assigning 
to  the  abscissa  x  any  series  of  values  and  computing  from  the 
equation  the  corresponding  values  of  the  ordinates.  This 
process  is  known  as  plotting  the  curve  by  points  ;  it  is  some- 
what laborious;  but  a  study  of  the  nature  of  the  quadratic 
function  will  show  that  the  determination  of  a  few  points  is 
sufficient  to  give  a  good  idea  of  the  curve. 

71 


7^ 


PLANE  ANALYTIC  GEOMETRY 


[V,  §  67 


Fig.  42 


67.  The  Form  y  =  ax\     Let  us  first  take  6  =  0,  c  =  0 ;  the 
resulting  equation 

(1)  y  =  cia? 

represents  a  parabola  which  passes  through  the  origin,  since 
the  values  0,  0  satisfy  the  equation.  This  parabola  is  symmet- 
ric with  respect  to  the  axis  Oy  ;  for,  the  values  of  y  correspond- 
ing to  any  two  equal  and  opposite  values  of  x  are  equal.  This 
line  of  symmetry  is  called  the  axis  of  the 
parabola ;  its  intersection  with  the  parab- 
ola is  called  the  vertex. 

We  may  distinguish  two  cases  accord- 
ing as  a  >  0  or  a  <  0 ;  if  a  =  0,  the  equa- 
tion becomes  y  =  0,  which  represents  the 
axis  Ox. 

(1)  If  a  >  0,  the  curve  lies  above  the  axis  Ox.  For,  no  matter 
what  positive  or  negative  value  is  assigned  to  a;,  y  is  positive. 
Furthermore,  as  x  is  allowed  to  increase  in  absolute  value,  y 
also  increases  indefinitely.  Hence  the  parabola  lies  in  the  first 
and  second  quadrants  with  its  vertex  at 
the  origin  and  opens  upward,  i.e.  is  con- 
cave upward  (Fig.  42). 

(2)  If  a  <  0,  we  conclude,  similarly, 
that  the  parabola  lies  below  the  axis  Ox, 
in  the  third  and  fourth  quadrants,  with 
its  vertex  at  the  origin  and  opens  down- 
wardf  i.e.  is  concave  downward  (Fig.  43). 

Draw  the  following  parabolas : 

y  =  x',y  =  Sx'',y  =  -\y?,y=\x'. 

68.  The  General  Equation.     The  curve  represented  by  the 
more  general  equation 

(2)  y  =  aa?  -{■hx-\-c 

differs  from  the  parabola  y  =  aa?  only  in  position.     To  see  this 


Fig.  43 


V,  §  69] 


THE  PARABOLA 


73 


we  use  the  process  of  completing  the  square  in  a;;   i.e.  we 
write  the  equation  in  the  equivalent  form 


=  K'^^J 


4a 


+  c; 


I.e. 


If  we  put 

2a  4a 


Fig.  44 


the  equation  becomes 

and  it  is  clear  (§  13)  that,  with  reference  to  parallel  axes 
OiXij  Oi2/i  through  the  point  Oi  (h,  k)  the  equation  of  the 
curve  is  2/1  =  <^^i  (Fig.  44).  The  parabola  (2)  has  therefore 
the  same  shape  as  the  parabola  y  =  ax^ ;  but  its  vertex  lies  at 
the  point  (h,  k),  and  its  axis  is  the  line  x  =  h.  The  curve 
opens  upward  or  downward  according  as  a  >  0  or  a  <  0. 

69.  Nature  of  the  Curve.  To  sketch  the  parabola  (2) 
roughly,  it  is  often  sufficient  to  find  the  vertex  (by  completing 
the  square  in  x^  as  in  §  68,  and  the  intersections  with  the  axes. 
The  intercept  on  the  axis  Oy  is  obviously  equal  to  c.  The  in- 
tercepts on  the  axis  Ox  are  found  by  solving  the   quadratic 

equation 

aar^  +  6a;  +  c  =  0. 

We  have  thus  an  interesting  interpretation  of  the  roots  of  any 
quadratic  equation:  the  roots  of  aa;^ -f- 6a; -h c  =  0  are  the 
abscissas  of  the  points  at  which  the  parabola  (2)  intersects 
the  axis  Ox.  The  ordinate  of  the  vertex  of  the  parabola 
is  evidently  the  least  or  greatest  value  of  the  function 
y  —  ax^  +  hx -\- c  according  as  a  is  greater  or  less  than  zero. 


74  PLANE  ANALYTIC  GEOMETRY  [V,  §  69 

EXERCISES 

1.  With  respect  to  the  same  coordinate  axes  draw  the  curves  y  =  ax'^ 
for  a =2,  f,  1,  i,  0,  —  i,  —  1,  -  |,  —  2.  What  happens  to  the  parabola 
y  =  ox2  as  a  changes  ? 

2.  Determine  in  each  of  the  following  examples  the  value  of  a  so  that 
the  parabola  y  =  ax^  will  pass  through  the  given  point : 

(a)  (2,3).  (b)   (-4,  1).  (c)   (-2,  -2).  (d)  (3,  -4). 

3.  A  body  thrown  vertically  upward  in  a  vacuum  with  a  velocity  of  v 
feet  per  second  will  just  reach  a  height  of  h  feet  such  that  h  =  ^v^. 
Draw  the  curve  whose  ordinates  represent  the  height  as  a  function  of  the 
initial  velocity. 

(a)  With  what  velocity  must  a  ball  be  thrown  vertically  upward  to  rise 
to  a  height  of  100  ft.  ? 

(6)  How  high  will  a  bullet  rise  if  shot  vertically  upward  with  an  ini- 
tial velocity  of  800  ft.  per  sec. ,  the  resistance  of  the  air  being  neglected  ? 

4.  The  period  of  a  pendulum  of  length  I  (i.e.  the  time  of  a  small 
back  and  forth  swing)  is  7"=  2iry/l/g.  Take  g  =  S2  ft. /sec.  and  di-aw 
the  curve  whose  ordinates  represent  the  length  I  of  the  pendulum  as  a 
function  of  the  period  T. 

(a)  How  long  is  a  pendulum  that  beats  seconds  (i.e.  of  period  2  sec.)  ? 
(6)  How  long  is  a  pendulum  that  makes  one  swing  in  two  seconds  ? 
(c)  Find  the  period  of  a  pendulum  of  length  one  yard. 

6.  Draw  the  following  parabolas  and  find  their  vertices  and  axes : 
(a)  y  =  \x^-x  +  2.         (b)  y  =  -  \  x^ -h  x.        (c)  y  =  5x^  +  16x  +  3. 
(d)y  =  2-x-x2,  (e)   y=x2-9.  (f)y  =  -9^x^. 

(fir)  y  =3x2- 6x -1-5.      (A)  y  =  ix2-|-2x-6.         (i)  x^  -  2x -y  =  0. 

6.  What  is  the  value  of  b  if  the  parabola  y  =  x^  -\-bx  —  6  passes 
through  the  point  (1,  6)  ?  of  c  if  the  parabola  y  =  x2  —  6x-|-c  passes 
through  the  same  point  ? 

7.  Suppose  the  parabola  y  =  ax^  drawn ;  how  would  you  draw  y  = 
a(x-|-2)2?  y  =  a(x-7)2?  y  =  ax'^  +  2?  y  =  ax2  -  7  ?  y  =  ax2-f  2x -|- 3  ? 

8.  What  happens  to  the  parabola  y  =  ax^  ■}-  bx -\-  c  as  'c  changes  ? 
For  example,  take  the  parabola  y  =  x2  —  x  +  c,  where  c  =  —  3,  —  2,  —  1, 
0,  1,  2,  3. 


V,  §69]  THE  PARABOLA  75 

9.    What  happens  to  the  parabola  y  =  ax^  -\-hx  -^  c  as  a  changes  ? 
For  example,  take  y  =  ax'^  —  x  —  Q,  where  a  =  2,  1,  ^,  0,-^,-1,  —  2. 

10.  (a)  If  the  parabola  y  =  ax^  +  bx  is  to  pass  through  the  points 
(1,  4),  (—2,  1)  what  must  be  the  values  of  a  and  b  ?  (&)  Determine  the 
parabola  y  =  ax^  +  bx -\- c  so  as  to  pass  through  the  points  (1,  ^),  (3,  2), 
(4,  f )  ;  sketch. 

11.  The  path  of  a  projectile  in  a  vacuum  is  a  parabola  with  vertical 
axis,  opening  downward.  With  the  starting  point  of  the  projectile  as 
origin  and  the  axis  Ox  horizontal,  the  equation  of  the  path  must  be  of  the 
form  y  =  ax^  +  bx.  If  the  projectile  is  observed  to  pass  through  the  points 
(30,  20)  and  (50,  30),  what  is  the  equation  of  the  path?  What  is  the 
highest  point  reached  ?    Where  will  the  projectile  reach  the  ground  ? 

12.  Find  the  equations  of  the  parabolas  determined  by  the  following 
conditions : 

(a)  the  axis  coincides  with  Oy,  the  vertex  is  at  the  origin,  and  the 
curve  passes  through  the  point  (—  2,  —  3)  ; 

(&)  the  axis  is  the  line  x  =  3,  the  vertex  is  at  (3,  —  2),  and  the  curve 
passes  through  the  origin  ; 

(c)  the  axis  is  the  line  x  =—  4,  the  vertex  is  (—  4,  6),  and  the  curve 
passes  through  the  point  (1,  2). 

13.  Sketch  the  following  parabolas  and  lines  and  find  the  coordinates 
of  their  points  of  intersection  : 

(a)  y  =  6x^,y  =  7x  +  S.  (6)  y -2x'^ ^Sx,  y  =  x  +  6. 

(c)  y  =  2-Sx^,y  =  2x  +  S.  (^a)  y  =  3 -\-x- x^,  x  +  y -  4  =  0. 

14.  Sketch  the  following  curves  and  find  their  intersections : 

(a)  x^  +  y^  =  25,  y  =  f  a;2.  (6)  3^^  +y^  ^  6y  =  0,  y  =  ^x^  -  2x  +  6. 

15.  The  ordinate  of  every  point  of  the  line  y  =  |  ic  +  4  is  the  sum  of 
the  corresponding  ordinates  of  the  lines  y  =  ^x  and  y  =  4.  Draw  the  last 
two  lines  and  from  them  construct  the  first  line. 

16.  The  ordinate  of  every  point  of  the  parabola  y  ^^x"^  +  ^x—l  is 
the  sum  of  the  corresponding  ordinates  of  the  parabola  y  =  ^x^  and  the 
line  y  =  ^x  —  1.    From  this  fact  draw  the  former  parabola. 

17.  The  ordinate  of  every  point  of  the  parabola  y  =  ^x^  —  x  +  Sis  the 
difference  of  the  corresponding  ordinates  of  the  parabola  y  =  ^x^  and  the 
line  y  =  x  —  S.    In  this  way  sketch  the  former  parabola. 


76  PLANE  ANALYTIC  GEOMETRY  [V,  §  70 

70.  Symmetry.  Two  points  P, ,  Pg  are  said  to  be  situated 
symmetncally  with  respect  to  a  line  Z,  if  /  is  the  perpendicular 
bisector  of  P^P^ ;  this  is  also  expressed  by  saying  that  either 
point  is  the  reflection  of  the  other  in  the  line  h 

Any  two  plane  figures  are  said  to  be  symmetric  with  respect 
to  a  line  I  in  their  plane  if  either  figure  is  formed  of  the  reflec- 
tions in  I  of  all  the  points  of  the  other  figure.  Each  figure  is 
then  the  reflection  of  the  other  in  the  line  I.  Two  such  figures 
are  evidently  brought  to  coincidence  by  turning  either  figure 
about  the  line  I  through  two  right  angles.  Thus,  the  lines 
y  =  2x-\-S  and  y  =  —  2x  —  3  are  symmetric  with  respect  to 
the  axis  Ox. 

A  line  I  is  called  an  axi's  of  symmetry  (or  simply  an  axis)  of 
a  figure  if  the  portion  of  the  figure  on  one  side  of  /  is  the 
reflection  in  Z  of  the  portion  on  the  other  side.  Thus,  any 
diameter  of  a  circle  is  an  axis  of  symmetry  of  the  circle. 
What  are  the  axes  of  symmetry  of  a  square  ?  of  a  rectangle  ? 
of  a  parallelogram? 

In  analytic  geometry,  symmetry  with  respect  to  the  axes  of 
coordinates,  and  to  the  lines  y=  ±x,  is  of  particular  importance. 

It  is  readily  seen  that  if  a  figure  is  symmetric  with  respect 
to  both  axes  of  coordinates,  it  is  symmetric  with  respect  to  the 
origiuy  i.e.  to  every  point  Pi  of  the  figure  there  exists  another 
point  P2ot  the  figure  such  that  the  origin' bisects  PiP2-  A 
point  of  symmetry  of  a  figure  is  also  called  center  of  the 
figure. 

EXERCISES 

1.  Give  the  coordinates  of  the  reflection  of  the  point  (a,  6)  in  the 
axis  Ox  ;  in  the  axis  Oy;  in  the  line  y  =  z  ;  in  the  line  y  =  2x  ;  in  the 
line  y  =—x. 

2.  Show  that  when  x  is  replaced  by  —  jc  in  the  equation  of  a  given  curve, 
we  obtain  the  equation  of  the  reflection  of  the  given  curve  in  the  y-axis. 


V,  §  70]  THE  PARABOLA  77 

3.  Show  that  when  x  and  y  are  replaced  by  y  and  x,  respectively,  in 
the  equation  of  a  given  curve,  we  obtain  the  equation  of  the  reflection  of 
the  given  curve  in  the  line  y  =  z. 

4.  Sketch  the  lines  y  =  -  2  ac  +  5  and  a;  =  -  2  y  +  5  and  find  their 
point  of  intersection. 

6.   Sketch  the  parabolas  y  =  a^  and  x  =  y^  and  find  their  points  of 

intersection. 

6.  Find  the  equation  of  the  reflection  of  the  line  2  a;  —  3  y  +  4  =  0  in 
the  line y  =  x\  in  the  axis  Ox ;  in  the  axis  Oy  ;  in  the  line  y-  —  x. 

7.  What  is  the  reflection  of  the  line  a;  =  a  in  the  line  y  =  a;  ?  in  the 
axes? 

8.  Find  and  sketch  the  circle  which  is  the  reflection  of  the  circle 
x2  +  y2  _  3  X  -  2  =  0  in  the  line  y  =  a;,  and  find  the  points  in  which  the 
two  circles  intersect. 

9.  Find  the  circle  which  is  the  reflection  of  the  circle  a;2-}-y2  —  4x4-3 
=  0  in  the  line  y  =  x\  in  the  coordinate  axes.  Sketch  all  of  these 
circles. 

10.  What  is  the  general  equation  of  a  circle  which  is  its  own  reflection 
in  the  line  y  =  a:  ?  in  the  axis  Ox  ?  in  the  axis  Oy  ?  What  circle  is  its 
own  reflection  in  all  three  of  these  lines  ? 

11.  What  is  the  equation  of  the  reflection  of  the  parabola  y  =— x2+4 
in  the  line  y  =  x  ?  in  the  line  y  =—  x  ?    Are  these  reflections  parabolas  ? 

12.  What  is  the  reflection  of  the  parabola  y=3x2  —  5x  +  6  in  the 
axis  Ox  ?  In  the  axis  Oy  ?    Are  these  reflections  parabolas  ? 

13.  If  the  cartesian  equation  of  a  curve  is  not  changed  when  x  is  re- 
placed by  —  X,  the  curve  is  symmetric  with  respect  to  Oy ;  if  it  is  not 
changed  when  y  is  replaced  by  —  y,  the  curve  is  symmetric  with  respect 
to  Ox ;  if  it  is  not  changed  when  x  and  y  are  replaced  by  —  x  and  —  y, 
respectively,  the  curve  is  symmetric  with  respect  to  the  origin ;  if  it  is 
not  changed  when  x  and  y  are  interchanged,  the  curve  is  symmetric  with 
respect  to  y  =  x. 


78 


PLANE  ANALYTIC  GEOMETRY 


[V,  §  71 


71.   Slope  of   Secant.     Let  P(x,  y)   be  any   point   of  the 
parabola 

(1)  y  =  a^. 
If  Pi  {xi,  yi)be  any  other  point  of 
this  parabola  so  that 

(2)  yi  =  axi\ 
the  line  PPj  (Fig.  45)  is  called  a 
secant. 

For  the  slope  tan  a^  of  this  secant 
we  have  from  Fig.  45 : 

(3) 


tan„.  =  ^'  =  -^ 


y^^y 


QQi 


Ax' 


or,  substituting  for  y  and  yi  their  values : 
(4) 


tan  «!  =  ^  W  -  a?')  ^  „/^  _j_  ^  X 
ari  —  X 


72.  Slope  of  Tangent.  Keeping  the  point  P  (Fig.  45) 
fixed,  let  the  point  P^  move  along  the  parabola  toward  P;  the 
limiting  position  which  the  secant  PPi  assumes  at  the  instant 
when  Pi  passes  through  P  is  called  the  tangent  to  the  parabola 
at  the  point  P. 

Let  us  determine  the  slope  tan  a  of  this  tangent.  As  the 
secant  turns  about  P  approaching  the  tangent,  the  point  Qi  ap- 
proaches the  point  Q,  and  in  the  limit  OQi  =  Xi  becomes  OQ=x. 
The  last  formula  of  §  71  gives  therefore  tan  a  if  we  make 
Xi  =  x:  tan  a  =  2  ax. 

The  slope  of  the  tangent  at  P  which  indicates  the  "steep- 
ness "  of  the  curve  at  P  is  also  called  the  slope  of  the  parabola 
at  P.  Thus  the  slope  of  the  parabola  y  =  ax^  at  any  point 
whose  abscissa  is  a;  is  =  2  ox ;  notice  that  it  varies  from  point 
to  point,  being  a  function  of  a?,  while  the  slope  of  a  straight 
line  is  constant  all  along  the  line. 


V,  §  73]  THE  PARABOLA  79 

The  knowledge  of  the  slope  of  a  curve  is  of  great  assistance 
in  sketching  the  curve  because  it  enables  us,  after  locating 
a  number  of  points,  to  draw  the  tangent  at  each  point.  Thus, 
for  the  parabola  y  =  |  ic^  ^^  find  tan  a=^x  -,  locate  the  points 
for  which  a;  =  0, 1,  2,  —  1,  —  2,  and  draw  the  tangents  at  these 
points ;  then  sketch  in  the  curve. 

73.  Derivative.  If  we  think  of  the  ordinate  of  the  parab- 
ola y  =  aa^  as  representing  the  function  ax^,  the  slope  of  the 
parabola  represents  the  rate  at  which  the  function  varies  with 
X  and  is  called  the  derivative  of  the  function  ax\  We  shall 
denote  the  derivative  of  y  by  y'.  In  §  72  we  have  proved  that 
the  derivative  of  the  function  y  =  ax^  is  y'  =  2  ax. 

The  process  of  finding  the  derivative  of  a  function,  which  is 
called  differentiation,  consists,  according  to  §§  71-72,  in  the 
following  steps:  Starting  with  the  value  y=ax^  of  the  func- 
tion for  some  particular  value  of  x  (say,  at  the  point  P,  Fig.  45), 
we  give  to  x  an  increment  Xi^—x  =  Ax  (compare  § 9)  and 
calculate  the  value  of  the  corresponding  increment  yi--y  =  Ay 
of  the  function.  Then  the  derivative  y'  of  the  function  y  is  the 
limit  that  Ay  /  Ax  approaches  as  Ax  approaches  zero.  In  the 
case  of  the  function  y  =  ax^  we  have 
Ay=y^-y  =  a{x^  -  a;^)  =  a\_{x  -h  Axf  -  x^]  =  a[2  xAx  +  {Axf] ; 

hence  -^  ==  a(2  a;  +  Ax). 

The  limit  of  the  right-hand  member  as  Ax  approaches  zero 

gives  the  derivative : 

?/'  =  2  ax. 

Thus,  the  area  y  of  a  circle  in  terms  of  its  radius  xiay  =  ttx^.  Hence 
the  derivative  y',  that  is  the  slope  of  the  tangent  to  the  curve  that  rep- 
resents the  equation  y  =  irx^,  is  y'  =  2Trx.  This  represents  (§72)  the 
rate  of  increase  of  the  area  y  with  respect  to  x.     Since  2  wx  is  the  length 


80  PLANE  ANALYTIC  GEOMETRY  [V,  §  73 

of  the  circumference,  we  see  that  the  rate  of  increase  of  the  area  y 
with  respect  to  the  radius  x  is  equal  to  the  circumference  of  the  circle. 

74.  Derivative  of  General  Quadratic  Function.  By  this 
process  we  can  at  once  find  the  derivative  of  the  general  quad- 
ratic function  y  =  aoi? -{- hx -\-  c  (§  66),  and  hence  the  slope  of 
the  parabola  represented  by  this  equation.    We  have  here 

Ay  =  a{x  +  Aa;)2  +  h{x  +  Aa;)  +  c  —  {ao?  -\-hx-{-c) 
=  2  ax^x  +  a(Aa;)2  -}-  h^x ; 

hence       — ^  =  2  aa;  +  6  +  a Aa;. 
Ax 

The  limit,  as  Aa;  becomes  zero,  is  2ax-\-h\  hence  the  deriva- 
tive of  the  quadratic  function  y  =ax^  -\-hx  -\-  c  is  y^  =^2  ax  •\-  b. 

76.  Maximum  or  Minimum  Value.  It  follows  both  from 
the  definition  of  the  derivative  as  the  limit  of  Ay/ Ax  and  from 
its  geometrical  interpretation  as  the  slope,  tana,  of  the  curve 
that  if,  for  any  value  of  x,  the  derivative  is  positive,  the  function, 
i.e.  the  ordinate  of  the  curve,  is  (algebraically)  increasing;  if 
the  derivative  is  negative,  the  function  is  decreasing. 

At  a  point  where  the  derivative  is  zero  the  tangent  to  the  curve 
is  parallel  to  the  axis  Ox.  The  abscissas  of  the  points  at  which 
the  tangent  is  parallel  to  Ox  can  therefore  be  found  by  equat- 
ing the  derivative  to  zero. 

In  this  way  we  find  that  the  abscissa  of  the  vertex  of  the 
parabola  y=aa?  -{-bx  -^  cis  x  =  —  6/2  a,  which  agrees  with  §  68. 

We  know  (§  68)  that  the  parabola  y  =  ax^ -\- bx -{- c  opens 
upward  or  downward  according  as  a  is  >  0  or  <  0.  Hence  the 
ordinate  of  the  vertex  is  a  minimum  ordinate,  i.e.  algebraically 
less  than  the  immediately  preceding  and  following  ordinates,  if 
a  >  0  ;  it  is  a  maximum  ordinate,  i.e.  algebraically  greater  than 
the  immediately  preceding  and  following  ordinates,  if  a  <  0. 

This  enables  us  to  determine  the  maximum  or  minimum  of 


V,  §  75]  THE  PARABOLA  81 

a  quadratic  function  ax^  +  &«  -f-  c  ;  the  value  of  x  for  which  the 
function  becomes  greatest  or  least  is  found  by  equating  the 
derivative  to  zero ;  the  quadratic  function  is  a  maximum  or  a 
minimum  for  this  value  of  x  according  as  a  <  0  or  >  0. 

Thus,  to  determine  the  greatest  rectangular  area  that  can  be  inclosed 
by  a  boundary  {e.g.  a  fence)  of  given  length  2  A;,  let  one  side  of  the 
rectangle  be  called  x ;  then  the  other  side  m  k  —  x.  Hence  the  area  A  of 
the  rectangle  \b  A  =  x{k  —  x)  =kx  —  x'^. 

Consequently  the  derivative  of  A  is  k—  2 x.  If  we  set  this  equal  to 
zero,  we  have  2x  =  kj  whence  x  =  k/2.  It  follows  that  k—x=k/2; 
hence  the  rectangle  of  greatest  area  is  a  square. 

EXERCISES 

1.  Locate  the  points  of  the  parabola  y  =  x^—4:X  +  ^  whose  abscissas 
are  --  1,  0,  1,  2,  3,  4,  draw  the  tangents  at  these  points,  and  then  sketch 
in  the  curve. 

2.  Sketch  the  parabolas  4:y=  —  x^  +  4:X  and  y  =  x^  —  S  by  locating 
the  vertex  and  the  intersections  with  Ox  and  drawing  the  tangents  at 
these  points. 

3.  Is  the  function  y  =  6(x^  —  4x  +  3)  increasing  or  decreasing  as  x 
increases  from  x  =  I?    from  x  =  | ? 

4.  Find  the  least  or  greatest  value  of  the  quadratic  functions : 
{a)2x^-Sx  +  6.         (&)  8-6x-a;2.  (c)x^-5x-6. 
(d)2-2a;-x2.            (g)4  +  x-^x2.  (/)  5x2  -  20  a;  +  1. 
6.  Find  the  derivative  of  the  linear  function  y  =  mx  +  b. 

6.  The  curve  of  a  railroad  track  is  represented  by  the  equation 
y  =  I  x2,  the  axes  Ox,  Oy  pointing  east  and  north,  respectively  ;  in  what 
direction  is  the  train  going  at  the  points  whose  abscissas  are  0,  1,2,  —  ^  ? 

7.  A  projectile  describes  the  parabola  y  =  fx—Zx^,  the  unit  being  the 
mile.  What  is  the  angle  of  elevation  of  the  gun  ?  What  is  the  greatest 
height  ?    Where  does  the  projectile  strike  the  ground  ? 

8.  A  rectangular  area  is  to  be  inclosed  on  three  sides,  the  fourth  side 
being  bounded  by  a  straight  river.  If  the  length  of  the  fence  is  a  con- 
stant k,  what  is  the  maximum  area  of  the  rectangle  ? 

Q 


82 


PLANE  ANALYTIC  GEOMETRY 


[V,§76 


Fio.  46 


4   . 

18  . 


PART   11.     POLYNOMIALS 

76.  The  Cubic  Function.  A  function 
of  the  form  a^a^  +  OiX^+azX  -|-  ttg  is  called  a 
cubic  function  of  x.  The  curve  repre- 
sented by  the  equation 

y  =0^  +  a^x^  +  a^x  +  a^ 
can  be  sketched  by  plotting  it  by  points 
(§  66). 

For  example,  to  draw  the  curve  repre- 
sented by  the  equation 

y  =  a;3-2a^-6a;-f  6, 
we  select  a  number  of  values  of  x  and  com- 
pute the  corresponding  values  of  y : 

a;=-3-2-101         23 

2/=- 24  0  860-40 
These  points  can  then  be  plotted  and  connected  by  a  smooth 
curve  which  will  approximately  represent  the  curve  corre- 
sponding to  the  given  equation  (Fig.  46). 

77.  Derivative.  The  sketching  of  such  a  cubic  curve  is 
again  greatly  facilitated  by  finding  the  derivative  of  the  cubic 
function ;  the  determination  of  a  few  points,  with  their  tan- 
gents, will  suffice  to  give  a  good  general  idea  of  the  curve. 

To  find  the  derivative  of  the  function  y  =  a^  +  a^a?  +  a^ 
-f  as  the  process  of  §  73  should  be  followed.     The  student 
may  carry  this  out  himself ;  he  will  find  the  quadratic  function 
2/'  =  3  a^"^  -I-  2  OiX  +  aj. 

78.  Maximum  or  Minimiun  Values.  The  abscissas  of 
those  points  of  the  curve  at  which  the  tangent  is  parallel  to 
the  axis  Ox  are  again  found  by  equating  the  derivative  to 
zero ;  they  are  therefore  the  roots  of  the  quadratic  equation 


V,  §80]  CUBIC  FUNCTIONS  83 

3  tto^  +  2  aiX  +  tta  =  0. 
If  at  such  a  point  the  derivative  passes  from  positive  to  nega- 
tive values,  the  curve  is  concave  doimiward,  and  the  ordinate 
is  a  maximum;  if  the  derivative  passes  from  negative  to  posi- 
tive values,  the  curve  is  concave  upward^  and  the  ordinate  is 
a  minimum. 

79.  Second  Derivative.  The  derivative  of  a  function  of 
X  is  in  general  again  a  function  of  x.  Thus  for  the  cubic 
function  y  =  aocc^  -\-  a^pi?  -{•  a>^  +  a^  the  derivative  is  the  quad- 
ratic function  2/'  =  3  a^^  +  2  a,a;  4-  «2. 

The  derivative  of  the  first  derivative  is  called  the  second  deriva- 
tive of  the  original  function ;  denoting  it  by  y'\  we  find  (§  74) 

2/"  =  6  aoo;  +  2  a^. 
As  a  positive  derivative  indicates  an  increasing  function, 
while  a  negative  derivative  indicates  a  decreasing  function 
(§  75),  it  follows  that  if  at  any  point  of  the  curve  the  second 
derivative  is  positive,  the  first  derivative,  i.e.  the  slope  of  the 
curve,  increases ;  geometrically  this  evidently  means  that  the 
curve  there  is  concave  upward.  Similarly,  if  the  second  de- 
rivative is  negative,  the  curve  is  concave  downward.  We  have 
thus  a  simple  means  of  telling  whether  at  any  particular  point 
the  curve  is  concave  upward  or  downward. 

It  follows  that  at  any  point  where  the  first  derivative  van- 
ishes, the  ordinate  is  a  minimum  if  the  second  derivative  is 
positive ;  it  is  a  maximum  if  the  second  derivative  is  negative. 

80.  Points  of  Inflexion.  A  point  at  which  the  curve 
changes  from  being  concave  downward  to  being  concave  up- 
ward, or  vice  versa,  is  called  a  point  of  inflexion.  At  such  a 
point  the  second  derivative  vanishes. 

Our  cubic  curve  obviously  has  but  one  point  of  inflection, 
viz.  the  point  whose  abscissa  is  a;  =  —  ai/(3  Oq). 


84  PLANE  ANALYTIC  GEOMETRY  [V,  §  80 

EXERCISES 

1.  Find  the  first  and  second  derivatives  of  y  v^hen : 

(a)       y  =  6 x8  -  7  a;2 - X  +  2.         (6)     y  =  20 -\- ix- Sx^  -  oi*. 
(c)  l0y  =  a^-5x^  +  Sx  +  9.         (d)     y  =  (x  -  l)(x- 2)(x-3). 
(e)       2/  =  x2(x+3).  (/)  7  2/  =  3x-2x(x2-l). 

2.  Sketch  the  curve  y  =  (x—  2)(x  +  1)  (x  +  3),  observing  the  sign  of  y 
between  the  intersections  with  Ox,  and  determining  the  minimum,  maxi- 
mum, and  point  of  inflection. 

3.  In  the  curve  y  =  aox^  +  aix^  +  azx  +  as,  what  is  the  meaning  of  as  ? 

4.  Sketch  the  curves  : 

(a)  6  y  =  (X  -  1)  (X  -f  iy.  (&)     y  =  (x  -  3)8. 

(c)  6  y  =  6  +  X  +  x'*  -  «*.  (d)     y  =  x«  -  4  X. 

(e)  8  y  =  5  x*  -  x8.  (/)     y  =  x«  -  3  x*  +  4  x  -  5. 

6.  Draw  the  curves  y  =  x,  y  =  x^,  y  =  x«,  with  their  tangents  at  the 
points  whose  abscissas  are  1  and  —  1. 

6.  Find  the  equation  of  the  tangent  to  the  curve  14  y  =  6  x*  —  2  x^ 
+x  —  20  at  the  point  whose  abscissa  is  2. 

7.  At  what  points  of  the  curve  y  =  x'  —  6  x^  +  3  are  the  tangents 
parallel  to  the  line  y=— 3x4-6? 

8.  Are  the  following  curves  concave  upward  or  downward  at  the 
indicated  points  ?     Sketch  each  of  them. 

(a)  y  =  4x8-6x,  atx  =  3.  (6)     3y  =  5x- 3x8,  at  x  =- 2. 

(c)  y  =  x8-2x2  + 6,  atx  =  ^.       (d)     2y  =  x8 -3x2,  at  x  =  1. 

Ce)  y  =  1 -x-x8,  atx  =  0.  (/)  10y=x8+x2-15x-!-6,atx=-|. 

9.  Show  that  the  parabola  y  =  ox*  +  6x  -f  c  is  concave  upward  or 
concave  downward  for  all  values  of  x  according  as  a  is  positive  or  negative. 

10.  The  angle  between  two  curves  at  a  point  of  intersection  is  the 
angle  between  their  tangents.  Find  the  angles  between  the  curves  y  =  x^ 
and  y  =  x8  at  their  points  of  intersection. 

11.  Find  the  angle  at  which  the  parabola  y  =  2x2  —  3x  —  6  intersects 
the  curve  y  =  x8  +  3  x  —  17  at  the  point  (2,  —  3). 

18.  The  ordinate  of  every  point  of  the  curve  y  =  x8  +  2  x*  is  the  sum  of 
the  ordinates  of  the  curves  y  =  x8  and  y  =  2  x^.  From  the  latter  two 
curves  construct  the  former. 


V,  §  81]  POLYNOMIALS  85 

13.  From  the  curve  y  =  x^  construct  the  following  curves  : 

(a)y  =  4x8.  (6)2/  =  ^|y.        (c)y  =  x3-.2.         (d)y  =  2a:8+4. 

14.  Draw  the  curve  2  ^  =  ic^  —  3  x^  and  its  reflection  in  the  line  y  =  x. 
What  is  the  equation  of  this  reflected  curve  ?  What  is  the  equation  of  the 
reflection  in  the  axis  Oy  ? 

15.  A  piece  of  cardboard  18  inches  square  is  used  to  make  a  box  by- 
cutting  equal  squares  from  the  four  corners  and  turning  up  the  sides. 
Draw  the  curve  whose  ordinates  represent  the  volume  of  the  box  as  a 
function  of  the  side  of  the  square  cut  out.     Find  its  maximum. 

16.  The  strength  of  a  rectangular  beam  cut  from  a  log  one  foot  in 
diameter  is  proportional  to  (i.e.  a  constant  times)  the  width  and  the 
square  of  the  depth.  Find  the  dimensions  of  the  strongest  beam  which 
can  be  cut  from  the  log.  Draw  the  curve  whose  ordinates  represent  the 
strength  of  the  beam  as  a  function  of  the  width. 

17.  Find  the  equation  of  the  curve  in  the  form  y  =  ax^  +  bx^  +  cx  +  d 
which  passes  through  the  following  points  : 

(a)  (0,  0),   (2,  -  1),    (-  1,  4),    (3,  4)  ; 
(6)  (1,  1),    (3,-1),    (0,  5),    (-4,1). 

18.  Show  that  every  cubic  curve  of  the  form  y  =  aoofi  +  aix^  +  azx  +  as 
is  symmetric  with  respect  to  its  point  of  inflection. 

81.  Polynomials.  The  methods  used  in  studying  the  quad- 
ratic and  cubic  functions  and  the  curves  represented  by  them 
can  readily  be  extended  to  the  general  case  of  the  polynomial, 
or  rational  integral  function,  of  the  nth  degree, 

y  =  a^x^  +  a^x^-^  -f  a^x^~''  +  -  +  (^n-v^  +  ^n? 
where  the  coefficients  a,,,  a^,  •••  a„  may  be  any  real  numbers, 
while  the  exponent  n,  which  is  called  the  degree  of  the  poly- 
nomial, is  a  positive  integer. 

We  shall  often  denote  such  a  polynomial  by  the  letter  y  or 
by  the  symbol  f{x)  (read :  function  of  x,  or  /  of  a;) ;  its  value 
for  any  particular  value  of  x,  say  a;  =  iCi  or  a;  =  ^,  is  then  de- 
noted by  f(x^  or  f(h),  respectively.  Thus,  for  a;  =  0  we  have 
/(0)  =  a„. 


86  PLANE  ANALYTIC  GEOMETRY  [V,  §  82 

82.  Calculation  of  Values  of  a  Polynomial.  In  plotting 
the  curve  y  =f(x)  by  points  (§§  66,  76)  we  have  to  calculate 
a  number  of  ordinates.  Unless  f(x)  is  a  very  simple  poly- 
nomial this  is  a  rather  laborious  process.  To  shorten  it  ob- 
serve that  the  value  f{x{)  of  the  polynomial 

f{x)  =  0^  +  ttiaf*-^  H f-  a„ 

f or  a;  =  a^  can  be  written  in  the  form 

/(^)  =  (  -  (((«oa?i  +  «i)»i  +  «2)a5i  +  cis)xi  -f  ...  +  a,_,)xi  +  a„. 

To  calculate  this  expression  begin  by  finding  a^pCi  -\-  a^ ;  mul- 
tiply by  Xi  and  add  aj ;  multiply  the  result  by  Xi  and  add  a^ ; 
etc.     This  is  best  carried  out  in  the  following  form : 


«o 

«i 

Oj 

..., 

aoX,+ 

«! 

( Vi  +  < 

!ll)Xi 

{a^i  +  < 

2y)Xi   - 

f  o^... 

instance, 

if 

/W  = 

=  2ar 

«-3a;2- 

-12a 

^  +  5 

=  ((2 

jc  -  3)a;  - 

-12); 

K4-5, 

to  find  /(3)  write  the  coefficients  in  a  row  and  place  2x3  =  6 
below  the  second  coefficient ;  the  sum  is  3.  Place  3  x  3  =  9  be- 
low the  third  coefficient ;  the  sum  is  —  3.  Place  3x(— 3)=— 9 
below  the  last  coefficient;  the  sum,  —4,  is  =/(3). 

2-3-12         6 

6  9-9 

2         3-3-4 

This  process  is  useful  in  calculating  the  values  of?/  that  cor- 
respond to  various  values  of  x,  as  we  have  to  do  in  plotting  a 
curve  by  points. 


V,  §  83] 


POLYNOMIALS 


87 


EXERCISES 

1.  If  /(x)=5x8-13x  +  2,  what  is  meant  by  /(a)?  hj  f{x-\-h)? 
What  is  the  value  of /(O)?  of/(2)?  of/(-3/5)?  of/(-l)? 

2.  Find  the  ordinates  of  the  curve  y=x^  — oi^  +  Sx'^  — 12x-\-S  for 
a;  =  3,  -9,  -^ 

3.  Find  the  ordinates  ot2y  =  x^  +  Sx^-20x-251otx  =  1,2,3,-  1,-2. 

4.  Suppose  the  curve  y  =/(a;)drawn  ;  how  would  you  sketch  : 
{a)y=f(x-2)?    ib)y=f(x+S)?    {c)y=f{2x)2    (d)y=f(-x)? 

(e)  y=f(^iy        (/)  y=fix)+5?   (g)  y=f(x)-2x? 

83.  Derivative  of  the  Polynomial.  We  have  seen  in  the 
preceding  sections  how  greatly  the  sketching  of  a  curve  and 
the  investigation  of  a  function  is  facilitated  by  the  use  of  the 
derivatives  of  the  function.  Thus,  in  particular,  the  first 
derivative  y'  is  the  rate  of  change  of  the  function  y  with  a;, 
and  hence  determines  the  slope,  or  steepness,  of  the  curve 
y  =f{x).  We  begin  therefore  the  study  of  the  polynomial  by 
determining  its  derivative.  The  method  is  essentially  the 
same  as  that  used  in  §  §  73,  74  for  finding  the  derivative  of 
a  quadratic  function. 

The  first  derivative  y'  of  any  function  y  of  a;  is  defined,  as 
in  §  73,  to  be  the  limit  of  the  quotient  Ay/^x  as  Ax  approaches 
zero,  Ay  being  the  increment  of 
the  function  y  corresponding  to 
the  increment  Ax  otx;  in  symbols : 

Sr'  =  liin^. 

Ax=0  Ax 


Geometrically  this  means   that  y' 
is  the  slope  of  the  tangent  of  the 


V 

" 

^/7  1                   1 

^ 

/\\                  \ 

*. 

y    0 

A 

/    i  «                 .Qi 

FiQ.  47 


curve  whose  ordinate  is  y. 
PP,  (Fig.  47)  : 


For,  Ay /Ax  is  the  slope  of  the  secant 


88  PLANE  ANALYTIC  GEOMETRY  [V,  §  83 

— ^  =  tan  «i ; 

Ax 

and  the  limit  of  this  quotient  as  Ax  approaches  zero,  i.e.  as  Pi 
moves  along  the  curve  to  P,  is  the  slope  of  the  tangent  at  P : 

y^  =  tan  a  =  lim^. 

Ax=oAa; 

If  the  function  y  be  denoted  by  /(«),  then 

Ay=f{x  +  Ax)-f(x)', 
hence 

Ax=o  Ax 

84.  Calculation  of  the  Derivative.  To  find,  by  means  of 
the  last  formula,  the  derivative  of  the  polynomial 

y  =/(aj)=  V  +  ai«""*  +  -  +  a.> 
we  should  have  to  form  first /(a;  -|-  Ax),  i.e. 

(x  +  Axy  -\-ai{x-\-  Axy-^  +  ...  +  a„, 
subtract  from  this  the  original  polynomial,  then  divide  by  Aa?, 
and  finally  put  Ax  =  0. 

This  rather  cumbersome  process  can  be  avoided  if  we 
observe  that  a  polynomial  is  a  sum  of  terms  of  the  form  ax* 
and  apply  the  following  fundamental  propositions  about 
derivatives : 

(1)  the  derivative  of  a  sum  of  terms  is  the  sum  of  the  derivor 
tives  of  the  terms  ; 

(2)  the  derivative  of  ax''  is  a  times  the  derivative  ofx""; 

(3)  the  derivative  of  a  constant  is  zero; 

(4)  the  derivative  of  x"  is  rix"~^ 

The  first  three  of  these  propositions  can  be  regarded  as 
obvious ;  a  fuller  discussion  of  them,  based  on  an  exact  defi- 
nition of  the  limit  of  a  function,  is  given  in  the  differential 


V,  §  85]  POLYNOMIALS  89 

calculus.     A  proof  of  the  fourth  proposition  is  given  in  the 
next  article. 

On  the  basis  of  these  propositions  we  find  at  once  that  the 
derivative  of  the  polynomial 

y  =  a^""  +  QiX""-^  -h  aiX""-"^  4-  -  +  a„_ia;  +  a„ 
is 

?/'  =  aowx^-i  -f  ai(n  —  l)a;"-2  _j_  ^2(72  —  2)x''-^  +  —  +  ««-!• 

85.   Derivative  of  x^.     By  the  definition  of  the  derivative 
(§83)  we  have  for  the  derivative  of  2/  =  x" : 

Ax=0  Ax 

Now  by  the  binomial  theorem  we  have 

{x  +  ^xY  =  a;"  +  na;"-iAa;  +  '*^^^^~^x''-\/^xy  +  —  +(Aa;)", 

JL  •  ^ 

and  hence 

{x  +  Aa;)'»  —  aj**  =  na;"-^Ax  +  ^^^  ~    -^  a;"-'^(Aa;)'^  +  ...  +  (Aa;)'*. 

Dividing  by  Aa;  and  then  letting  Aa;  become  zero,  we  find 
y'  =  na;""^ 

EXERCISES 

1.  Find  the  derivatives  of  the  following  functions  of  x  by  means  of 
the  fundamental  definition  (§  83)  and  check  by  §  84  : 

(a)  a;8.  (6)  x^  +  x.  (c)  x*  +  6  x^. 

(d)  -  6  a;8.  (c)  x*  -  S  x^.  (J)  mx  +  b. 

2.  Find  the  derivatives  of  the  following  functions : 

(a)  5  a;*- 3x2  +  6  X.    (b)  1-x+l  x^-^x^       (c)    (x  -  2)3. 

(d)  (2  X  +  3)5.  (e)  3(4  x  -  l)^.  (/)  x'^  +  ax'»-i+  ftx'-s. 

3.  For  the  following  functions  write  the  derivative  indicated  : 
(a)  6  x8  -  3  X,  find  y'".  (6)    ax^  +  bx  +  c,  find  y'". 

(c)  x6,  find^.  W   ax3  +  ftx2  +  ex  +  (?,  find  yiv. 

(e)  ix6,findy".  (/)  i^x«,  find  i,vii. 

(g)  x^  -  gx8,  find  y'".  {h)    (2  x  -  3)3,  find  y'". 


90  PLANE  ANALYTIC  GEOMETRY  [V,  §  86 

86.  Properties  of  the  General  Polynomial  Curve.  In  plot- 
ting the  curve 

y  =  Ooaj*  +  aiaf~^  -f  a.^"'^  -{-  ...  -{-a^ 

observe  that  (Fig.  48) : 

(a)  the  intercept  OB  on  the  axis  Oy 
is  equal  to  the  constant  term  a„ ; 

(6)  the  intercepts  OA^  OA2,  ••.  on 
the  axis  Ox  are  roots  of  the  equation 
y  =  0,  i.e.  '    FiQ.  48 

aoX"H-ai.'B"-iH-  ...  +a,  =  0; 

(c)  the  abscissas  of  the  least  and  greatest  ordinates  are 
found  by  solving  the  equation  y'  =  0,  i.e.  (§  84) 

every  real  root  giving  a  minimum  ordinate  if  for  this  root  y" 
is  positive  and  a  maximum  ordinate  if  y"  is  negative ; 

(d)  the  abscissas  of  the  points  of  inflection  are  found  by 
solving  the  equation  y"  =  0,  i.e. 

7i(n-l)aoX''-2+  —  +2a„_s  =  0, 

every  real  root  of  this  equation  being  the  abscissa  of  a  point 
of  inflection  provided  that  y'"=^0.  (If  y"'  were  zero,  y'  might 
not  be  a  maximum  or  minimum,  and  further  investigation 
would  be  necessary.) 

87.  Continuity  of  Poljnaomials.  It  should  also  be  ob- 
served that  the  function  y  =  a^pf"  +  Ojaf ~^  +  —  -\-  a„  is  one- 
valued,  real,  and  finite  for  every  x ;  i.e.  to  every  real  and  finite 
abscissa  x  belongs  one  and  only  one  ordinate,  and  this  ordinate  is 
real  and  finite.  Moreover,  as  the  first  derivative  y'  =  nooa:""^ 
-{-•••  +a„_i  is  again  a  polynomial,  the  slope  of  the  curve  is 
everywhere  one-valued  and  finite. 


V,  §  88] 


POLYNOMIALS 


91 


Thus,  so-called  discontinuities  of  the  ordinate  (Fig.  49)  or  of 

the  slope  (Fig.  50)  cannot  occur :  the  curve  y  =  a^x""  -\ h  a„ 

is  continuous. 


Strictly  defined,  the  continuity  of  the  function  y  =  Ooic"  +  ••• 
4-  a„  means  that,  for  every  value  of  x,  the  limit  of  the  function 
is  equal  to  the  value  of  the  function.  The  function  y  =  aoCC*  +  — 
+  a„  has  one  and  only  one  value  for  any  value  x  =  Xi,  viz. 
a^j^  +  ...  +  a^.  The  value  of  •  the  function  for  any  other 
value  of  X,  say  for  a^  +  ^x,  is  a^ix^  +  Aa;)"  -}-  ••.  -f  a^  which  can 
be  written  in  the  form  ao^i"  +  —  +  «n  +  terms  containing  Ax 
as  factor.  Therefore  as  Ax  approaches  zero,  the  function 
approaches  a  limit,  viz.  its  value  for  x  =  Xi. 

88.  Intermediate  Values.  A  continuous  function,  in 
varying  from  any  value  to  any  other  value,  must  necessarily 
pass  through  all  intermediate  values.  Thus,  our  polynomial 
y  =  aoX""  +  —  -f  a„,  if  it  passes  from  a  negative  to  a  positive 
value  (or  vice  versa) ^  must  pass  through  zero.  It  follows 
from  this  that  beticeen  any  two  ordinates  of  opposite  sign  the 
curve  y  =  cTo^"  +  •••  +  <^u  ''^ust  cross  the  axis  Ox  at  least  once. 

It  also  follows  from  the  continuity  of  the  polynomial  and 
its  derivatives  that  between  any  two  intersections  with  the  axis 
Ox  there  must  lie  at  least  one  maximum  or  minimum^  and  be- 
tween a  maximum  and  a  minimum  there  must  lie  a  point  of 
inflection. 

Ordinates  at  particular  points  can  be  calculated  by  the  pro- 
cess of  §  82. 


92  PLANE  ANALYTIC  GEOMETRY  [V,  §  88 

EXERCISES 
1.   Sketch  the  following  curves  : 
(a)  y=(a:-l)(x-2)(x-3).       (&)  iy  =  x^-l.  (c)  10y  =  x^. 

{d)  10y  =  x5  +  5.  (e)  4y  =  (a;+2)2(a;-3).     (/)  y={x-iy. 

8.  When  is  the  curve  y  =  aox"  +  aix^-^  +  •••  +  an  symmetric  with 
respect  to  Oy? 

3.  Determine  the  coefficients  so  that  the  curve  y  =  oqx^  +  aix*  +  a^^^ 
+  asx  +  a4  shall  touch  Ox  at  (1,  0)  and  at  (  —  1,  0)  and  pass  through 
(0,  1),  and  sketch  the  curve. 

4.  Find  the  coordinates  of  the  maxima,  minima,  and  points  of  inflec- 
tion and  then  sketch  the  curve  4  y  =  x*  —  2  x^. 

5.  Are  the  following  curves  concave  upward  or  downward  at  the  indi- 
cated points  ? 

(a)  16y=16x*-8x2  +  l,  atx=-l,  -J,  0,  i,   3. 
(6)       y  =  4x-x*,  atx=:-2,  0,  1,  3. 

(c)  y  =  x",  at  any  point ;  distinguish  the  cases  when  n  is  a  positive 
even  or  odd  integer. 

6.  What  happens  to  the  curves  y  =  ox*  and  y  =  aX^  as  a  changes  ? 
For  example,  take  a  =2,  1,  ^,  0,  —  |,  —  1,  —  2. 

7.  Find  the  values  of  z  for  which  the  following  relations  are  true : 
(a)  X*- 6x2 +  9^0.  (6)  (x  -  l)2(x2- 4)  ^0. 

8.  Those  curves  whose  ordinates  represent  the  values  of  the  first, 
second,  etc.,  derivatives  of  a  given  polynomial  are  called  the  first,  second, 
etc.,  derived  curves.  Sketch  on  the  same  coordinate  axes  the  following 
curves  and  their  derived  curves  : 

(a)  6y  =  2x«-3x2-12x.  (6)  y  =(x  -  2)2(x  +  1). 

(c)      y=(x-|-l)8.  (d)  2y  =  x*  4x2  +  1. 

9.  At  what  point  on  Ox  must  the  origin  be  taken  in  order  that  the 
equation  of  the  curve  y  =  2x'— 3x2  —  12x  —  5  shall  have  no  term  in  x*  ? 
no  term  in  x  ? 

10.  Find,  to  three  significant  figures,  the  roots  of  the  equation 
x8-3x+l=0. 


CHAPTER  VI 


THE   PARABOLA 


89.  The  Parabola.  The  parabola  can  be  defined  as  the 
locus  of  a  point  whose  distance  from  a  fixed  point  is  equal  to  its 
distance  from  a  fixed  line.  The  fixed  point  is  called  the  focus, 
the  fixed  line  the  directrix,  of  the  parabola. 

Let  F  (Fig.  51)  be  the  fixed  point,  d  the  fixed  linej  then 

every  point  P  of  the  parabola  must  satisfy 

the  condition 

FP=PQ, 

Q  being  the  foot  of  the  perpendicular  from 
P  to  d.  Let  us  take  F  as  origin,  or  pole,  and 
the  perpendicular  FD  from  F  to  the  directrix 
as  polar  axis,  and  let  the  given  distance  FD 
=  2  a.  Then  FP  =r  2ind  PQ  =  2  a -r  cos  cf>. 
The  condition  PP=PQ  becomes  therefore 

^^  r  =  2  a  —  r  cos  <^, 

(1)  r=      ^^      . 

^  ^  1  +  cos  <j> 

This  equation,  which  expresses  the  radius  vector  of  P  as  a 
function  of  the  vectorial  angle  <j>,  is  the  polar  equation  of  the 
parabola,  when  the  focus  is  taken  as  pole  and  the  perpendicular 
from  the  focus  to  the  directrix  as  polar  axis. 

90.  Polar  Construction  of  Parabolas.  By  means  of  the 
equation  (1)  the  parabola  can  be  plotted  by  points.  Thus,  for 
<^  =  0  we  find  r  =  a  as  intercept  on  the  polar  axis.  As  <^ 
increases  from  the  value  0,  r  continually  increases,  reaching 

93 


Fig.  51 


94 


PLANE  ANALYTIC  GEOMETRY 


[VI,  §  90 


the  value  2  a  for   <^  =  i7r,  and  becoming  infinite  as   <f>  ap- 
proaches the  value  tt. 

For  any  negative  value  of  <^  (between  0  and  —  tt)  the  radius 
vector  has  the  same  length  as  for  the  corresponding  positive 
value  of  <}> ;  this  means  that  the  parabola  is  symmetric  with 
respect  to  the  polar  axis. 

The  intersection  A  of  the  curve  with  its  axis  of  symmetry 
is  called  the  vertex,  and  the  axis  of 
symmetry  FA  the  axis,  of  the  parab- 
ola. The  segment  BB'  cut  off  by 
the  parabola  on  the  perpendicular  to 
the  axis  drawn  through  the  focus  is 
called  the  latus  rectum;  its  length 
is  4  a,  if  2  a  is  the  distance  between 
focus  and  directrix.  Notice  also  that 
the  vertex  A  bisects  this  distance 
FD  so  that  the  distance  between  focus 
and  vertex  as  well  as  that  between  vertex  and  directrix  is  a. 

In  Fig.  62  the  polar  axis  is  taken  positive  in  the  sense  from 
the  pole  toward  the  directrix.  If  the  sense  from  the  directrix 
to  the  pole  is  taken  as  positive  (Fig.  52),  we  have  again  with 
-Fas  pole  FP=r,  but  the  distance  of  P  from  the  directrix  is 
2  a  +  ?•  cos  <f>,  So  that  the  polar  equation  is  now 

(2)  r  =  -^^~ 

^  ^  1  -  cos  <^ 

We  have  assumed  a  as  a  positive  number,  2  a  denoting  the 
absolute  value  of  the  distance  between  the  fixed  point  (focus) 
and  the  fixed  line  (directrix).  The  radius  vector  r  is  then 
always  positive.  But  the  equations  (1)  and  (2)  still  represent 
parabolas  if  a  is  a  negative  number,  viz.  (1)  the  parabola  of 
Fig.  52,  (2)  the  parabola  of  Fig.  51,  the  radius  vector  r  being 
negative  (§  16). 


Fig.  52 


VI,  §  92] 


THE  PARABOLA 


95 


<f 

w 

ifc* 

f^n 

^^/  * 

• 

^  / 

X     / 

/     / 

A ' 

J) 

alajf 

d 

V 

Fig.  53 


91.  Mechanical  Construction.  A  mechanism  for  traciag 
an  arc  of  a  parabola  consists  of  a  right- 
angled  triangle  (shaded  in  Fig.  53),  one  of 
whose  sides  is  applied  to  the  directrix. 
At  a  point  It  of  the  other  side  HQ  3l 
string  of  length  MQ  is  attached ;  the  other 
end  of  the  string  is  attached  at  the  focus 
F.  As  the  triangle  slides  along  the  di- 
rectrix, the  string  is  kept  taut  by  means 
of  a  pencil  at  P  which  traces  the  parabola. 
Of  course,  only  a  portion  of  the  parabola  can  thus  be  traced, 
since  the  curve  extends  to  infinity. 

92.  Transformation  to  Cartesian  Coordinates.  To  obtain 
the  cartesian  equation  of  the  parabola  let  the  origin  0  be  taken 
at  the  vertex,  i.e.  midway  between  the  fixed  line  and  fixed 
point,  and  the  axis  Ox  along  the  axis  of  the  parabola,  positive 
in  the  sense  from  vertex  to  focus  (Fig.  54).  Then  the  focus 
F  has  the  coordinates  a,  0,  and  the  equation  of  the  directrix  is 
x  =  —a.  The  distance  FP  of  any  point 
P(x,  y)  of  the  parabola  from  the  focus  is 
therefore  V(a;  —  a)^  -f- 1/^,  and  the  dis- 
tance QP  of  P  from  the  directrix  is 
a-\-x.     Hence  the  equation  is 

(x-ay  +  y''=(a-\-xy, 
which  reduces  at  once  to 
(3)  2/2  =  4  ax. 

This  then  is  the  cartesian  equation  of  the  parabola,  referred 
to  vertex  and  axis,  I.e.  when  the  vertex  is  taken  as  origin  and 
the  axis  of  the  parabola  (from  vertex  toward  focus)  as  axis  Ox. 

Notice  that  the  ordinate  at  the  focus  (a,  0)  is  of  length  2  a ; 
the  double  ordinate  B'B  at  the  focus  is  the  latus  rectum  (§  90). 


Fig.  54 


96 


PLANE  ANALYTIC  GEOMETRY 


[VI,  §  93 


93.  Negative  Values  of  a.  In  the  last  article  the  constant 
a  was  again  regarded  as  positive ;  but  (compare  §  90)  the  equa- 
tion (3)  still  represents  a  parabola  when  a  is  a  negative  number, 
the  only  difference  being  that  in  this  case  the  parabola  turns  its 
opening  in  the  negative  sense  of  the  axis  Ox  (toward  the  left 
in  Fig.  54).  Thus  the  parabolas  2/^=4  ax  and  y^=  —  4  aa  are  sym- 
metric to  each  other  with  respect  to  the  axis  Oy  (Ex.  14,  p.  77). 

The  equation  (3)  is  very  convenient  for  plotting  a  parabola 
by  points.  Sketch,  with  respect  to  the  same  axes,  the  parab- 
olas :  y^  =  16xjy^  =  — 16  x,  y^  =  x,  y^  =  —  x,  y^=Sx,  y'^  =  — \  x, 

94.  Axis  Vertical.     The  equation 
(4)  x'=4:ayy 

which  differs  from  (3)  merely  by  the  interchange  of  x  and  y, 
evidently  represents  a  parabola  whose  vertex  lies  at  the  origin 
and  whose  axis  coincides  with  the  axis  Oy.  The  parabolas  (3) 
and  (4)  are  each  the  reflection  of  the  other  in  the  line  y  =x 
(Ex.  14,  p.  77).     The  equation  (4)  can  be  written  in  the  form 

y  =  ^x^. 
4a 

As  1/4  a  may  be  any  constant,  this  is  the  equation  discussed  in 
§  67. 


95.   New  Origin.     An  equation  of  the  form  (Fig.  55) 


(5) 


(y  —  ky  =  4:  a(x 


y 


JtA 


Q 

Fig.  55 


k\ 


Q 

FiQ.  56 


or  of  the  form  (Fig.  56) 

(6)  (x-^)2  =  4a(2/-A;), 


VI,  §96]  THE  PARABOLA  97 

evidently  represents  a  parabola  whose  vertex  is  the  point  (^,  k), 
while  the  axis  is  in  the  former  case  parallel  to  Ox,  in  the  latter 
to  Oy.  For,  by  taking  the  point  (h,  k)  as  new  origin  we  can 
reduce  these  equations  to  the  forms  (3),  (4),  respectively. 

The  parabola  (5)  turns  its  opening  to  the  right  or  left,  the 
parabola  (6)  upward  or  downward,  according  as  4  a  is  positive 
or  negative. 

96.  General  Equation.  The  equations  (5),  (6)  as  well  as 
the  equations  (3),  (4)  are  of  the  second  degree.  Now  the 
general  equation  of  the  second  degree  (§  47), 

Ax"  +  2  Hxy  -^  Bf  +  2  Ox  +  2  Fy  +  C  =  0, 
can  be  reduced  to  one  of  the  forms  (5),  (6)  if  it  contains  no 
term  in  xy  and  only  one  of  the  terms  in  x^  and  2/S  i-^-  if  H  =  0 
and  either  ^  or  ^  is  =0.  This  reduction  is  performed  (as  in 
§  48)  by  completing  the  square  my  ov  x  according  as  the  equa- 
tion contains  the  term  in  y^  or  x\ 

Thus  any  equation  of  the  second  degree,  containing  no  term  in 
xy  and  only  one  of  the  squares  x"^,  y^,  represents  a  parabola,  whose 
vertex  is  found  by  completing  the  square  and  whose  axis  is 
parallel  to  one  of  the  axes  of  coordinates. 

EXERCISES 

1.  Sketch  the  following  parabolas : 

(a)  r  = , (6)  r  =  -—^ (c)  r  =  asec^  ^  0. 

^  ^         1  +  cos  0  ^  ^         1  -  cos  0 

2.  Sketch  the  following  curves  and  find  their  intersections  : 

2  a 

(a)  r  =  8  cos  <l>,  r  = (6)  r  =  a,  r  = 


1  —  cos  0  1  +  cos  0 

8  2  (I 

(c)  r  =  4  cos  0,  r  = (d)  r  cos  0  =  2  a,  r  =  - • 

1  +  cos  0  1  -  cos  0 

3.   Sketch  the  following  parabolas  : 
(a)   (y_2)2  =  8(x~5).  (6)  (x  +  3)2  =  5(3  -  y). 

(c)  x2  =  6(y  +  1).  {d)  (2/  +  3)2  =  -  3  X. 


98  PLANE  ANALYTIC  GEOMETRY  [VI,  96 

4.   Sketch  each  of  the  following  parabolajs  and  find  the  coordinates  of 
the  vertex  and  focus,  and  the  equations  of  the  directrix  and  axis  : 
(a)  2/2-2y-3x- 2  =  0.  (6)  x2  +  4x- 4  y  =  0. 

(c)  a;2__4a;  +  3y +  1  =  0.  (d)  Sx^  -  6x  -  y  =  0. 

(e)   8  y2  -  16  y  4-  X  +  6  =  0.  (/)  y^  +  y  +  x  =  0. 

(^)  x2~x-3y  +  4  =  0.  (A)  8y2_3x  +  3=0. 

6.  Sketch  the  following  loci  and  find  their  intersections  : 
(a)  y  =  2x,  y  =x2.  (6)  y*  =  4ax,  x -f  y  =  3a. 

(c)  y2  =  x  +  3,  y2  =  5-x.  (d)  y2  4-4x  +  4=0,     x2+y2=41. 

6.  Sketch  the  parabolas  with  the  following  lines  and  points  as  direc- 
trices and  foci,  and  find  their  equations  : 

(a)  X  -  4  =  0,  (6,   -  2).  (6)  y  4-  3  =  0,  (0,  0). 

(c)  2x  +  6  =  0,  (0,  -1).  (d)  x  =  0,  (2,  -3). 

(e)  3y-l  =  0,  (-2,1).  (/)  x-2a  =  0,  (a,  6). 

7.  Find  the  parabola,  with  axis  parallel  to  Ox,  and  passing  through 
the  points : 

(a)  (1,  0),  (5,  4),  (10,  -6).  (6)  (V,  -5),  (f,  0),  (j,  -3). 

8.  Find  the  parabola,  with  axis  parallel  to  Oy^  and  passing  through 
the  points : 

(a)   (0,  0),  (-2,  1),  (6,  9).  (d)  (1,  4),   (4,  -1),  (-3,  20). 

9.  Find  the  parabola  whose  directrix  is  the  line  3x  —  4y  —  10=0  and 
whose  focus  is:  (a)  at  the  origin;  (6)  at  (6,  —  2).  Sketch  each  curve. 
When  does  the  equation  of  a  parabola  contain  an  xy  term  ? 

10.  Find  the  parabolas  with  the  following  points  as  vertices  and  foci 
(two  solutions)  : 

(a)  (-3,  2),  (-3,  6).  (6)  (2,  5),  (-1,  5). 

(c)    (-1,    -1),  (1,  -1).  {d)  (0,  0),  (0,  -a). 

11.  If  s  denotes  the  distance  (in  feet)  from  a  point  P  in  the  line  of 
motion  of  a  falling  body,  at  a  time  <  (in  seconds), 

s-so  =  ig(t-to)% 
where  g  is  the  gravitational  constant  (32.2  approximately)  and  Sq  is  the 
distance  from  F  at  the  time  ^q,  show  that  this  equation  can  be  put  in  the 
standard  form 

s  =  lgT, 


VI,  §  97] 


THE  PARABOLA 


99 


where  s  denotes  the  distance  from  some  other  fixed  point  in  the  line  of 
motion  and  7  is  the  time  since  the  body  was  at  that  point. 

12.  The  melting  point  t  (in  degrees  Centigrade)  of  an  alloy  of  lead  and 

zinc  is  found  to  be 

f  =  133  +  .875a;  +  .01125a;2, 

where  x  is  the  percentage  of  lead  in  the  alloy.  Reduce  the  equation  to 
standard  form  t  =  kx  ;  and  show  that  x  —  x  —  h^  t=  t  —  k,  where  h  is 
the  percentage  of  lead  that  gives  the  lowest  melting  point,  and  k  is  the 
temperature  at  which  that  alloy  melts. 

13.  Show  that  the  locus  of  the  center  of  the  circle  which  passes 
through  a  fixed  point  and  is  tangent  to  a  fixed  line  is  a  parabola. 

14.  Show  that  the  locus  of  the  center  of  a  circle  which  is  tangent  to  a 
fixed  line  and  a  fixed  circle  is  a  parabola.  Find  the  directrix  of  this 
parabola. 

97.   Slope  of  the  Parabola.     The  slope  tan  a  of  the  parabola 

at  any  point  P  (x,  y)  (Fig.  57)  can  be  found  (comp.  §  72)  by 
first  determining  the  slope 

tan «! = y^^y 

of  the  secant  PP^ ,  and  then  letting 

-Pi(2Jij  2/i)  move  along  the  curve  up 

to  the  point  P(x,   y).     Now   as   P^ 

conies  to  coincide  with  P,  x-^  becomes 

equal  to  x,  and  y^  equal  to  y,  so  that 

the  expression  for  tan  «!  loses  its 

meaning.    But  observing  that  P  and 

Pi  lie  on  the  parabola,  we  have  y^^^^ax  and  y^  =  4  aa^i ,  and 

hence  y^  —  y^  —  A^a{xi  —  x).      Substituting  from  this  relation 

the  value  of  x^  —  x  in  the  above  expression  for  tan  «!,  we  find 

for  the  slope  of  the  secant : 

tan  a^  =  ^a  -^ — '-  = 

Vi-y^     2/1  +  2/ 


Fig.  57 


100  PLANE  ANALYTIC  GEOMETRY  [VI,  §  97 

If  we  now  let  Pj  come  to  coincidence  with  P  so  that  yi  becomes 
=  y,  we  find  for  the  slope  of  the  tangent  at  P{x,  y)  : 

(7)  tana  =  ^. 

y 

This  slope  of  the  tangent  at  P  is  also  called  the  slope  of  the 
parabola  at  P.  The  ordinate  y  of  the  parabola  is  a  function  of 
the  abscissa  x ;  and  the  slope  of  the  parabola  at  P  (x,  y)  is  the 
rate  at  which  y  increases  with  increasing  x  at  P;  in  other  words, 
it  is  the  derivative  y'  of  y  with  respect  to  x  (compare  §  73). 

As  by  the  equation  of  the  parabola  we  have  y  =  ±  2  Va^,  we 
find: 

(8)  2/'  =  tana  =  ?^=±J^. 

y         ^x 

The  double  sign  in  the  last  expression  corresponds  to  the  fact 
that  to  a  given  value  of  x  belong  two  points  of  the  curve  with 
equal  and  opposite  slopes. 

98.   Equation  of  the  Tangent.     As  the  slope  of  the  parabola 
2/2  =4  oa; 

at  the  point  P(x,  y)  is  2  a/y  (§  97),  the  equation  of  the  tangent 
at  this  point  is 

Y-y=^{X-x), 

y 

where  X,  Fare  the  coordinates  of  any  point  of  the  tangent, 
while  Xj  y  are  the  coordinates  of  the  point  of  contact.  This 
equation  can  be  simplified  by  multiplying  both  sides  by  y  and 
observing  that  y*  =  4  aa; ;  we  thus  find 

(9)  yT=^2aix  +  X). 

Notice  that  (as  in  the  case  of  the  circle,  §  64)  the  equation 
of  the  tangent  is  obtained  from  the  equation  of  the  curve, 
y^  =  4:ax,  by  replacing  y^  hy  yT,  2  x  hy  x  -\-  X. 


VI,  §  99] 


THE  PARABOLA 


101 


The  segment  TP  (Fig.  58)  of  the  tangent  from  its  intersec- 
tion T  with  the  axis  of  the 
parabola  to  the  point  of  contact 
P  is  called  the  length  of  the 
tangent  at  P  ;  the  projection  TQ 
of  this  segment  TP  on  the  axis 
of  the  parabola  is  called  the 
subtangent  at  P.  Now,  with 
Y=  0,  equation  (9)  gives  X  =  ^  x,  i.e.  TO  =  OQ;  hence  the 
subtangent  is  bisected  by  the  vertex.  This  furnishes  a  simple 
construction  for  the  tangent  at  any  point  P  of  the  parabola  if 
the  axis  and  vertex  of  the  parabola  are  known. 


99.  Equation  of  the  Normal.  The  normal  at  a  point  P 
of  any  plane  curve  is  defined  as  the  perpendicular  to  the  tan- 
gent through  the  point  of  contact. 

The  slope  of  the  normal  is  therefore  (§  27)  minus  the  recip- 
rocal of  that  of  the  tangent.  Hence  the  equation  of  the  normal 
to  the  parabola  is  : 

Y 

that  is : 


,  =  -J^(X-.), 


(10) 


yX-^2aY=(2a-\-x)y. 


The  segment  PN  of  the  normal  from  the  point  P{x,  y) 
on  the  curve  to  the  intersection  N  of  the  normal  with  the  axis 
of  the  parabola  is  called  the  leyigth  of  the  normal  at  P;  the 
projection  QJSf  of  this  segment  PN  on  the  axis  of  the  parabola 
is  called  the  subnormal  at  P. 

Now,  with  F=0,  equation  (10)  gives  X  =  2  a-{-x,  and  as 
x=OQ,  it  follows  that  Q]Sf=2a;  i.e.  the  subnormal  of  the 
parabola  is  constant,  viz.  equal  to  half  the  latus  rectum. 


102  PLANE  ANALYTIC  GEOMETRY        [VI,  §  100 

100.  Intersections  of  a  Line  and  a  Parabola.  The  inter- 
sections of  the  parabola 

with  the  straight  line 

y  =  mx  +  h 

are  found  by  substituting  the  value  of  y  from  the  latter  in  the 
former  equation : 

{mx  -I-  6)2  =  4  ax, 
or,  reducing : 

m V  +  2(m6-2a)a;  +  62  =  0. 

The  roots  of   this  quadratic  in  x  are  the  abscissas  of  the 
points  of  intersection ;  the  ordinates  are  then  found  from 
y  =  mx  -4-  h. 

It  thus  appears  that  a  straight  line  cannot  intersect  a  parabola 
in  more  than  two  points.  If  the  roots  are  imaginary,  the  line 
does  not  meet  the  parabola ;  if  they  are  real  and  equal,  the 
line  has  but  one  point  in  common  with  the  parabola  and  is 
a  tangent  to  the  parabola  (provided  m  =^  0). 

101.  Slope  Equation  of  the  Tangent.    The  condition  for 

equal  roots  is 

(6m-2a)»  =  6»m», 
which  reduces  to 

The  point  that  the  line  of  this  slope  has  in  common  with  the 
parabola  is  then  found  to  have  the  coordinates 

2a  — bm      6*  _^  .  .      ok 

X  = =  — ,  y  =  ma;  -f  6  =  J  o. 

m*  a 

As  the  slope  of  the  parabola  at  any  point  (x,  y)  is  (§  97) 
2/  =  2  a/y,  the  slope  at  the  point  just  found  is  3/  =  ajb  —  m ; 
i.e.  the  slope  of  the  parabola  is  the  same  as  that  of  the  line 
y  =  mx-\-b\   this  line  is  therefore  a  tangent.     Thus,  the  line 


VI,  §  103]  THE  PARABOLA  103 

(11)  y=^mx+-  y 

is  tangent  to  the  parabola  y^  =  4  ax  whatever  the  value  of  m. 

This  may  be  called  the  slope-form  of  the  equation  of  the  tangent. 
Equation  (11)  can  also  be  deduced  from  the  equation  (9),  by 
putting  2  a/y  =  m  and  observing  that  2/^  =  4  aaj. 

102.  Slope  Equation  of  the  Normal.  The  equation  (10)  of 
the  normal  can  be  written  in  the  form 

2a  2a 

or  since  by  the  equation  (3)  of  the  parabola  x  =  y^/4:  a : 

2a      ^^^Sa' 
If  we  denote  by  n  the  slope  of  this  normal,  we  have : 

n  =  -^,  y  =  -2an,  ^^-an\ 
2  a  8  a^ 

so  that  the  equation  of  the  normal  assumes  the  form 

(12)  T=nX-2an-  an\ 

This  may  be  called  the  slopeform  of  the  equation  of  the  normal. 

103.  Tangents  from  an  Exterior  Point.  The  slope-form 
(11)  of  the  tangent  shows  that  from  any  point  (x,  y)  of  the  plane 
not  more  than  two  tangents  can  be  drawn  to  the  parabola  y'^  =  ^ax. 
For,  the  slopes  of  these  tangents  are  found  by  substituting  in 
(11)  for  X,  y  the  coordinates  of  the  given  point  and  solving  the 
resulting  quadratic  in  m.  This  quadratic  may  have  real  and 
different,  real  and  equal,  or  complex  roots. 

Those  points  of  the  plane  for  which  the  roots  are  real  and 
different  are  said  to  lie  outside  the  parabola ;  those  points  for 
which  the  roots  are  imaginary  are  said  to  lie  within  the  parab- 


104 


PLANE  ANALYTIC  GEOMETRY        [VI,  §  103 


ola;  those  points  for  which  the  roots  are  equal  lie  on  the 
parabola.     The  quadratic  in  m  can  be  written 

xm^  —  ym  -f  a  =  0, 

so  that  the  discriminant  is  y*  —  4  ax.  Therefore  a  point  (x^  y) 
of  the  plane  lies  within,  on,  or  outside  the  parabola  according  as 
2/*  —  4  oa;  is  less  than,  equal  to,  or  greater  than  zero. 

Similarly,  the  slope-form  (12)  of  the  normal  shows  that  not 
more  than  three  normals  can  be  drawn  from  any  point  of  the 
plane  to  the  parabola,  since  the  equation  (12)  is  a  cubic  for  n 
when  the  coordinates  of  any  point  of  the  plane  are  substituted 
for  X,  T.  As  a  cubic  has  always  at  least  on§  real  root  there 
always  exists  one  normal  through  a  given  point;  but  there 
may  be  two  or  three. 


104.  Geometric  Properties.  Let  the  tangent  and  normal 
at  P  (Fig.  59)  meet  the  axis  at  T,  N-,  let  Q  be  the  foot  of  the 
perpendicular  from  P  to 
the  axis,  D  that  of  the  per- 
pendicular to  the  directrix 
d;  and  let  0  be  the  vertex, 
F  the  focus. 

As  the  subtangent  TQ  is 
bisected  by  0  (§  98)  and 
the  subnormal  QN  is  equal 
to  2  a  (§  99),  while  OF  = 
a,  it  follows  that  F  lies 
midway  between  T  and  N. 

The  triangle  TPN  being 
right-angled  at  P,  and  F  being  the  midpoint  of  its  hypotenuse, 


it  follows  that 


FP=^FT=FN. 


VI,  §  105] 


THE  PARABOLA 


105 


Hence,  if  axis  and  focus  are  given,  the  tangent  and  the  normal 
at  any  point  P  of  the  parabola  are  found  by  describing  about 
F  a  circle  through  P  which  will  meet  the  axis  at  T  and  N. 

As  FP=DP,  it  follows  that  FPDT  is  a  rhombus;  the 
diagonals  PT  and  FD  bisect  therefore  the  angles  of  the 
rhombus  and  intersect  at  right  angles.  As  TP  (like  TQ)  is 
bisected  by  the  tangent  at  the  vertex,  the  intersection  of  these 
diagonals  lies  on  this  tangent  at  the  vertex.  The  properties 
just  proved  that  the  tangent  at  P  bisects  the  angle  between  the 
focal  radius  PF  and  the  parallel  PD  to  the  axis  and  that  the 
perpendicular  from  the  focus  to  the  tangent  meets  the  tangent  on 
the  tangent  at  the  vertex  are  of  particular  importance. 

105.  Diameters.  It  is  known  from  elementary  geometry  that 
in  a  circle  all  chords  parallel  to  any  given  direction  have  their 
midpoints  on  a  straight  line  which  is  a  diameter  of  the  circle. 

Similarly,  in  a  parabola,  the  locus  of  the  midpoints  of  all  chords 
parallel  to  any  given  direction  is  a  straight  line^  and  this  line 
which  is  parallel  to  the  axis 
is  called  a  diameter  of  the 
parabola.  To  prove  this,  take 
the  vertex  as  origin  and  the 
axis  of  the  parabola  as  axis  Ox 
(Fig.  60)  so  that  the  equation 
is  2/^^  =  4  ax.  Any  line  of  given 
slope  m  has  the  equation 

y  =  mx-\-b,  Fig.  60 

and  with  variable  b  this  represents  a  pencil  of  parallel  lines. 
Eliminating  x  we  find  for  y  the  quadratic 


106  PLANE  ANALYTIC  GEOMETRY        [VI,  §  105 

The  roots  ^j,  y^  are  the  ordinates  of  the  points  Pj,  Pg  at 
which  the  line  intersects  the  parabola.     The  sum  of  the  roots  is 

4a 

2/1  +  ^2  =  — ; 
m 

hence  the  ordinate  \{yi  +  2/2)  of  the  midpoint  P  between  Pj ,  Pj 
is  constant  (i.e.  independent  of  »),  viz.  =  2  a/m,  and  independ- 
ent of  6.  The  midpoints  of  all  chords  of  the  same  slope  m 
lie,  therefore,  on  a  parallel  to  the  axis,  at  the  distance  2  a/m 
from  it.  The  condition  for  equal  roots  (§  101)  gives  h  =  a/m. 
That  one  of  the  parallels  which  passes  through  the  point  where 
the  diameter  meets  the  parabola  is,  therefore, 

y  =  mx-Jf-—\ 
m 

by  §  101  this  is  a  tangent.  Thus,  the  tangent  at  the  end  of  a 
diameter  is  parallel  to  the  chords  bisected  by  the  diameter. 

EXERCISES 

1.  Find  and  sketch  the  tangent  and  normal  of  the  following  parabolas 
at  the  given  points  : 

(a)  2y2  =  26x,  (2,  5).        (6)   3y2  =  4x,  (3,  -  2).     (c)  y^  =  2x,  {\,  1). 
(d)  5y2=i2x,  (i-2).    (6)  y^  =  x,{hl).  (/)  46y«  =  x,  (5,  i). 

2.  Show  that  the  secant  through  the  points  P  (a; ,  y)  and  Pi  (xi ,  j/i) 
of  the  parabola  y^  =  iax  has  the  equation  4aX— (y  +  yi)  F+ yyi  =  0, 
and  that  this  reduces  to  the  tangent  at  P  when  Pi  and  P  coincide. 

3.  Find  the  angle  between  the  tangents  to  a  parabola  at  the  vertex 
and  at  the  end  of  the  latus  rectum.  Show  that  the  tangents  at  the  ends  of 
the  latus  rectum  are  at  right  angles. 

4.  Find  the  length  of  the  tangent,  subtangent,  normal,  and  subnormal 
of  the  parabola  y^  ^^  4  ^  at  the  point  (1,  2). 

6.  Find  and  sketch  the  tangents  to  the  parabola  y^  =  Sx  from  each 
of  the  following  points : 

(a)  (-  2,  3).  (6)  (-  2,  0).  (c)  (-6,  0).  (d)  (8,  8). 


VI,  §  105]  THE  PARABOLA  107 

6.  Draw  the  tangents  to  the  parabola  ?/2  =  3  x  that  are  inclined  to  the 
axis  Ox  at  the  angles:  (a)  30°,  (&)  45°,  (c)  135°,  (d)  150°;  and  find 
their  equations. 

7.  Find  and  sketch  the  tangents  to  the  parabola  y^  =  Ax  that  pass 
through  the  point  ( —  2,  2). 

8.  Find  and  sketch  the  normals  to  the  parabola  y^  =  6z  that  pass 
through  the  points  : 

(«)  (1,0).     (6)  (¥,-3).     (c)  (-*/,-!).     ((^)(f,-|).     (e)  (0,0). 

9.  Are  the  following  points  inside,  outside,  or  on  the  parabola 
Sy^  =  x?     (a)  (3,1).     (6)  (2,  i).     (c)  (8,  1).     (c?)  (10,  f). 

10.  Show  that  any  tangent  to  a  parabola  intersects  the  directrix  and 
latus  rectum  (produced)  in  points  equally  distant  from  the  focus. 

11.  Show  that  the  tangents  drawn  to  a  parabola  from  any  point  of  the 
directrix  are  perpendicular. 

12.  Show  that  the  ordinate  of  the  intersection  of  any  two  tangents  to 
the  parabola  y^  =  'iax  is  the  arithmetic  mean  of  the  ordinates  of  the 
points  of  contact,  and  the  abscissa  is  the  geometric  mean  of  the  abscissas 
of  the  points  of  contact. 

13.  Show  that  the  sum  of  the  slopes  of  any  two  tangents  of  the  parab- 
ola 2/2  =  4  aa:  is  equal  to  the  slope  Y/Xoi  the  radius  vector  of  the  point  of 
intersection  (X,  F)  of  the  tangents  ;  find  the  product  of  the  slopes. 

14.  Find  the  locus  of  the  intersection  of  two  tangents  to  the  parabola 
2/2  =  4  ax^  if  the  sum  of  the  slopes  of  the  tangents  is  constant. 

15.  Find  the  locus  of  the  intersection  of  two  perpendicular  tangents  to 
a  parabola  ;  of  two  perpendicular  normals  to  a  parabola. 

16.  Show  that  the  angle  between  any  two  tangents  to  a  parabola  is 
half  the  angle  between  the  focal  radii  of  the  points  of  contact. 

17.  From  the  vertex  of  a  parabola  any  two  perpendicular  lines  are 
drawn ;  show  that  the  line  joining  their  other  intersections  with  the 
parabola  cuts  the  axis  at  a  fixed  point. 

18.  Find  and  sketch  the  diameter  of  the  parabola  y"^  —  Qx  that  bisects 
the  chords  parallel  to  3a;  —  2y  +  5  =  0;  give  the  equation  of  the  focal 
chord  of  this  system. 

19.  Find  the  system  of  parallel  chords  of  the  parabola  y2  =  8  x  bisected 
by  the  line  y  =  3. 


108  PLANE  ANALYTIC  GEOMETRY        [VI,  §  105 

20.  Show  that  the  tangents  at  the  extremities  of  any  chord  of  a  parab- 
ola intersect  on  the  diameter  bisecting  this  chord.     Compare  Ex.  12. 

21.  Find  the  length  of  the  focal  chord  of  a  parabola  of  given  slope  m. 

22.  Find  the  angles  at  which  the  parabolas  y^  =  4ax  and  x^  =  4ay 
intersect. 

23.  Two  equal  confocal  parabolas  have  the  same  axis  but  open  in  op- 
posite sense  ;  show  that  they  intersect  at  right  angles. 

24.  If  axis,  vertex,  and  one  other  point  of  the  parabola  are  given,  ad- 
ditional points  can  be  constructed  as  follows  :  Let  0  be  the  vertex,  P  the 
given  point,  and  Q  the  foot  of  the  perpendicular  from  P  to  the  tangent 
at  the  vertex ;  divide  QP  into  equal  parts  by  the  points  ^i,  J.2,  •••  ;  and 
OQ  into  the  same  number  of  equal  parts  by  the  points  Pi,  P2,  •••  ;  the 
intersections  of  0-4i,  0^2»  •••  with  the  parallels  to  the  axis  through  Pi, 
P2,  •••are  points  of  the  parabola. 

26.  If  two  tangents  APu  AP^  to  a  parabola  with  their  points  of  con- 
tact Pi,  P2  are  given  and  ^Pi,  ^P2  be  divided  into  the  same  number  of 
equal  parts,  the  points  of  division  being  numbered  from  Pi  to  A  and  from 
A  to  P2,  the  lines  joining  the  points  bearing  equal  numbers  are  tangents 
to  the  parabola.  To  prove  this  show  that  the  intersections  of  any  tangent 
with  the  lines  ^Pi,  ^P2  divide  the  segments  Pi^,  APi  in  the  same 
division  ratio. 

26.  The  shape  assumed  by  a  uniform  chain  or  cable  suspended  between 
two  fixed  points  Pi,  P2  is  called  a  catenary  ;  its  equation  is  not  algebraic 
and  cannot  be  given  here.  But  when  the  line  P1P2  is  nearly  horizontal 
and  the  depth  of  the  lowest  point  below  P1P2  is  small  in  comparison  with 
P1P2,  the  catenary  agrees  very  nearly  with  a  parabola. 

The  distance  between  two  telegraph  poles  is  120  ft.  ;  P2  lies  2  ft.  above 
the  level  of  Pi ;  and  the  lowest  point  of  the  wire  is  at  1/3  the  distance  be- 
tween the  poles.  Find  the  equation  of  the  parabola  referred  to  Pi  as 
origin  and  the  horizontal  line  through  Pi  as  axis  Ox  ;  determine  the  posi- 
tion of  the  lowest  point  and  the  ordinates  at  intervals  of  20  ft. 

27.  The  cable  of  a  suspension  bridge  assumes  the  shape  of  a  parabola 
if  the  weight  of  the  suspended  roadbed  (together  with  that  of  the  cables) 
is  uniformly  distributed  horizontally.  Suppose  the  towers  of  a  bridge 
240  ft.  long  are  60  ft.  high  and  the  lowest  point  of  the  cables  is  20  ft.  above 
the  roadway  ;  find  the  vertical  distances  from  the  roadway  to  the  cables 
at  intervals  of  20  ft. 


VI,  §  106]  THE  PARABOLA  109 

28.  When  a  parabola  revolves  about  its  axis,  it  generates  a  surface  called 
a  paraboloid  of  revolution ;  all  meridian  sections  (sections  through  the 
axis)  are  equal  parabolas.  If  the  mirror  of  a  reflecting  telescope  is  such 
a  surface  (the  portion  about  the  vertex),  all  rays  of  light  falling  in  parallel 
to  the  axis  are  reflected  to  the  same  point ;  explain  why. 

106.  Parameter  Equations.  Instead  of  using  the  cartesian 
or  polar  equation  of  a  curve  it  is  often  more  convenient  to 
express  x  and  y  (or  r  and  <^)  each  in  terms  of  a  third  variable, 
which  is  then  called  the  parameter. 

Thus  the  parameter  equations  of  a  circle  of  radius  a  about  the 
origin  as  center  are  : 

x  =  a  cos  </>,     y  =  a  sin  <^, 

</)  being  the  parameter.  To  every  value  of  <f>  corresponds  a 
definite  x  and  a  definite  y,  and  hence  a  point  of  the  curve.- 
The  elimination  of  <^,  by  squaring  and  adding  the  equations, 
gives  the  cartesian  equation  a^-\-y^  =  a^. 

Again,  to  determine  the  motion  of  a  projectile  we  may  observe 
that,  if  gravity  were  not  acting,  the  projectile,  started  with  an 
initial  velocity  v^  at  an  angle  c  to  the  horizon,  would  have  at  the 
time  t  the  position 

a;  =  ^>o  cos  c  •  ^,     y  =  VQ  sin  e  •  t^ 

the  horizontal  as  well  as  the  vertical  motion  being  uniform. 
But,  owing  to  the  constant  acceleration  g  of  gravity  (down- 
ward), the  ordinate  y  is  diminished  by  ^gt"^  in  the  time  i,  so 
that  the  coordinates  of  the  projectile  at  the  time  t  are 
x  =  Vq cos  c  •  ^,     y  =  VQmie't  —  \gt'^. 

These  are  the  parameter  equations  of  the  path,  the  parameter 
here  being  the  time  t.  The  elimination  of  t  gives  the  cartesian 
equation  of  the  parabola  described  by  the  projectile : 

y=zVQtdiXi€'X-       /        x\ 
2  V  cos^  € 


110 


PLANE  ANALYTIC  GEOMETRY        [VI,  §  107 


107.  Parameter  Equations  of  a  Parabola.    For  any  parabola 

^  =  4  oa;  we  can  also  use  as  parameter  the  angle  a  made  by  the 

tangent  with  the  axis  Ox\  we  have  for  this  angle  (§  97) : 

2a 
tan  a  =  — ; 

y 

it  follows  that  y  =  2a  cot  a  and  hence  x  =  y^/4:  a  =  a  cot*  «. 

The  equations 

x  =  a  cot'  a,     y  =  2  a  cot  a 

are  parameter  equations  of  the  parabola  ?/'  =  4  oic ;  the  elimina- 
tion of  cot  a  gives  the  cartesian  equation. 

108.  Parabola    referred  to  Diameter  and  Tangent.    The 

equation  of  the  parabola  y^  =  4ax  preserves  this  simple  form  if  instead  of 
axis  and  tangent  at  the  vertex  we  take  as 
axes  any  diameter  and  the  tangent  at  its  end. 
We  shall  show  that   the  equation  in  these 
oblique  coordinates  is 

yi*  =  4  aiXi , 
where  oi  is  a  new  constant  determined  below. 
To  prove  this  observe  that  since  the  new 
origin    Oi  (A,  A;)  is  a  point  of  the  parabola 
y*  =  4  aXy  we  have  by  §  107 

h  =  acot^a,       k  =  2a cot  a,  ^o-  61 

where  a  is  the  angle  at  which  the  tangent  at  Oi  is  inclined  to  the  axis. 
Hence,  transferring  to  parallel  axes  through  Oi,  we  obtain  the  equation 

(y  +  2  a  cot  a)2  =  4  a  (x  +  a  cot2  a), 
which  reduces  to 

y^  -\-iacot(t'y  =  iax. 

The  relation  between  the  rectangular  coordinates  x,  y  and  the  oblique 
coordinates  Xi ,  yi ,  both  with  Oi  as  origin,  is  readily  seen  from  the  figure 
to  be  X  =  xi  +  yi  cos  a,  y  =  yi  sin  a.     Substituting  these  values  we  find 
2/1^  sin2  a  +  4  a  cos  a  •  yi  =  4  oici  +  4  ayi  cos  a, 


y 

/ 

h    /  \                            1          X 

/' 

or,  if  we  put  a/sin^  a  =  ai,  y^  —  4 


xi  =  4  a\X\. 


VI,  §  108]  THE  PARABOLA  111 

The  meaning  of  the  constant  ai  appears  by  observing  that 

sin2  a         tan2  a 
ai  is  therefore  the  distance  of  the  new  origin  Oi  from  the  directrix,  or, 
what  amounts  to  the  same,  from  the  focus  F. 


EXERCISES 

1.  Show  that  the  parameter  equations  of  a  circle  with  center  at  (A,  k) 

and  radius  a  are 

X  =  h  -i-  a  cos  4>,     y  =  k-{-  asiiKp. 

2.  Sketch  the  curves  whose  equations  are  : 

(a)  x  =  t,y  =  t'^;  (b)  x  =  t^ -1,  y  =  3-21"^; 

(c)  x  =  2t-l,  y  =  t^-St';      ((?)  X  =  3  +  2 cos  0,  2/  =  4  +  2 sin 0 ; 

(e)  a;  =  4  +  5  cos  0,  y  =  2  +  5  sin  0. 

3.  What  must  be  the  initial  velocity  vq  of  a  projectile  if,  with  an  eleva- 
tion of  30°,  it  is  to  strike  an  object  100  ft.  above  the  horizontal  plane  of 
starting  point  at  a  horizontal  distance  from  the  latter  of  1200  ft.? 

4.  What  must  be  the  elevation  e  to  strike  an  object  100  ft.  above  the 
horizontal  plane  of  the  starting  point  and  6000  ft.  distant,  if  the  initial 
velocity  be  1200  ft.  per  second  ? 

6.  Prove  that  a  projectile  whose  elevation  is  60°  rises  three  times  as 
high  as  when  its  elevation  is  30°,  the  magnitude  of  the  initial  velocity 
being  the  same  in  each  case. 

6.  If  a  golf  ball  be  driven  from  the  tee  horizontally  with  initial  speed 
=  300  ft. /sec,  where  and  when  would  it  land  on  ground  16  ft.  below  the 
tee  if  resistance  of  air  and  rotation  of  ball  could  be  neglected  ? 

7.  A  man  standing  15  feet  from  a  pole  150  ft.  high  aims  at  the  top  of 
the  pole.  If  the  bullet  just  misses  the  top,  where  will  it  strike  the  ground 
if  vo  -  1000  ft.  /sec.  ? 

8.  The  ends  ^,  J5  of  a  straight  rod  of  length  2  a  move  along  two  per- 
pendicular lines  ;  find  the  locus  of  the  midpoint  of  AB. 

9.  Four  rods  are  jointed  so  as  to  form  a  parallelogram  ;  if  one  side  is 
fixed,  find  the  path  described  by  any  point  rigidly  connected  with  the  op- 
posite side. 


112 


PLANE  ANALYTIC  GEOMETRY        [VI,  §  109 


109.  Area  of  Parabolic  Segment.  A  parabola,  together  with 
any  chord  perpendicular  to  its  axis,  bounds  an  area  OPF^  (shaded  in 
Fig.  62).  It  was  shown  by  Archimedes  (about 
250  B.C.)  that  this  area  is  two  thirds  the  area 
of  the  rectangle  PPfQ'Q  that  has  the  chord 
J" P  as  one  side  and  the  tangent  at  the  vertex 
as  opposite  side.  Fig.  62 

This  rectangle  PJPQ'Q  is  often  called  (somewhat  improperly)  the  cir- 
cumscribed rectangle  so  that  the  result  can  be  expressed  briefly  by  saying 
that  the  area  of  the  parabola  is  2/3  of  that  of  the  circumscribed  rectangle. 

This  statement  is  of  course  equivalent  to  saying  that  the  (non-shaded) 
area  OQP  is  1/3  of  the  area  of  the  rectangle  OQPB.  In  this  form  the 
proposition  is  proved  in  the  next  article. 

110.  Area  by  Approximation  Process.  To  obtain  first  an  ap- 
proximate value  {A)  for  the  area  OQP  (Fig.  63)  we  may  subdivide  the 
area  into  rectangular  strips  of  equal  width, 
by  dividing  OQ  into,  say,  n  equal  parts 
and  drawing  the  ordiuates  j/i ,  y2,  •••  ^n- 
If  the  width  of  these  strips  is  Ax  so  that 
OQ  =  nAx,  we  have  as  approximate  value 
of  the  area : 

(A)  =  Ax  •  yi  -\-  Ax '  7/2  +  •"  +  Ax  •  y„. 
Now  yi  is  the  ordinate  corresponding  to  the  abscissa  Ax  ;  yz  corresponds 
to  the  abscissa  2  Ax,  etc.  ;  yn  corresponds  to  the  abscissa  nAx  =  OQ. 
Hence,  if  the  equation  of  the  curve  is  a;'*  =  4  ay,  we  have  : 


FiQ.  63 


4a 


2/2 


T^(2Ax)2, 
4a 


yn  =  ^{nAxy. 
4a 


Substituting  these  values  we  find  : 


(^)=(M-'(i  +  22  +  32+  ... +n2). 
4  a 


Now, 


1  +  22+  ...  +n2  =  in(n+l)(2n+l)=i(2n8  +  3n2  +  n);- 

hence  (^)  =  ^'(2n«  +  3n2  +  n)  =  C^Y2 +  §  +  I). 

^    '      24  a  24  a   \        n     n^) 

Now  nAx  =  OQ  —  Xn,  the  abscissa  of  the  terminal  point  P,  whatever  the 
number  n  and  length  Ax  of  the  subdivisions.     Hence,  if  we  let  the  num- 


VI,  §  111] 


THE  PARABOLA 


113 


ber  n  increase  indefinitely,  we  find  in  the  limit  the  exact  expression  A  for 
the  area  OQP: 


._^_1. 


4a     3 


where  y„  =  Xr?/4,  a  is  the  ordinate  of  the  terminal  point  P.  As  XnVn  is 
the  area  of  the  rectangle  OQPB,  Archimedes'  proposition  (§  109)  is 
proved. 

111.  Area  expressed  in  Terms  of  Ordinates.  The  area 
(shaded  in  Fig.  64)  between  the  parabola  cc^  =  4  ay,  the  axis  Ox,  and  the 
two  ordinates  yi,  y^,  whose  abscissas  differ  by  y 
2  Ax  is  evidently,  by  the  formula  of  §  110, 


A  =  -^{x^^-Xi^). 
12  a 

Ax 


12  a 


=  -=^  (6  xi^  +  12  xiAx 
12  a 


[(xi  +  2  Ax)8-xi3] 
-  8  (Ax)2). 


Fig.  64 


This  expression  can  be  given  a  remarkably 
simple  form  by  introducing  not  only  the  ordinates  y\  =  a;iV4  a,  yz  = 
(ici  +  2  Axy/i  a,  but  also  the  ordinate  2/2  midway  between  yi  and  yz, 
whose  abscissa  is  Xi  +  Ax.    For  we  have  : 


yi  +  4  2/2  + 


=  ~[xi^-hHxi  +  Axy  +{xi  +  2  Ax)2] 
4a 


We  find  therefore : 


=  —  [6  Xi2  +  12  XiAx  +  8(Ax)2]. 
4/z 

^  =  |Ax(yi  +  4?/2  +  y3). 


This  formula  holds  even  if  the  vertex  of 
the  parabola  is  at  any  point  {h,  k) ,  pro- 
vided the  axis  of  the  parabola  is  parallel  to 
Oy.  For  (Fig.  65 ) ,  to  find  the  area  under 
the  arc  P1P2P3  we  have  only  to  add  to 
the  doubly  shaded  area  the  simply  shaded 
rectangle  whose  area  is  2  kAx.  We  find 
therefore  for  the  whole  area : 

I  Ax(yi  +  4  ?/2  +  2/3)  +  2  Mx  =  ^  Ax(yi  +  4 1/2  +  2/3  +  6  A:) 

=  1  Ax  l(yi  +  A;)  +  4  (2/2  +  k)  +(2/3  +  A:)], 

I 


Fio.  65 


114 


PLANE  ANALYTIC  GEOMETRY        [VI,  §  111 


Fig.  66 


where  yi,y2,  yz  are  the  ordinates  of  the  parabola  referred  to  its  vertex, 
and  hence  yi  +  k^  yi  +  k^  yz-\-  k  the  ordmates  for  the  origin  O. 
We  have  therefore  for  any  parabola  whose  axis  is  parallel  to  Oy  : 
A  =  \  ^x{yi  +  iy2-{-y3). 

112.  Approximation  to  any  Area.    Simpson's  Rule.    The 

last  formula  is  sometimes  used  to  find  an  approximate  value  for  the  area 

under  any  curve  (i.e.  the  area  bounded 

by  the  axis  Ox,  an  arc  AB  of  the  curve, 

and  the  ordinates  of  ^  and  B,  Fig.  66). 

This  method  is  particularly  convenient 

if  a  number  of  equidistant  ordinates 

of  the  curve  are  known,   or  can   be 

found  graphically. 

Let  Ax  be  the  distance  of  the  ordi- 
nates, and  let  yi ,  2/2 »  ys  be  any  three 
consecutive  ordinates.  Then  the  doubly  shaded  portion  of  the  required 
area,  between  yi  and  ys,  will  be  (if  Ax  is  sufficiently  small)  very  nearly 
equal  to  the  area  under  the  parabola  that  passes  through  Pi ,  P3 ,  P3  and 
has  its  axis  parallel  to  Oy.    This  parabolic  area  is  by  §  111 

=  ^Ax(yi+^yi  +  y3). 
The  whole  area  under  AB  is  a  sum  of  such  expressions.  This  method 
for  finding  an  approximate  expression  for  the  area  under  any  curve  is 
known  as  Simpson's  rule  (Thomas  Simpson,  1743)  although  the  funda- 
mental idea  of  replacing  an  arc  of  the  curve  by  a  parabolic  arc  had  been 
suggested  previously  by  Newton. 

113.  Area  of  any  Parabolic  Segment.  As  the  equation  of  a 
parabola  referred  to  any  diameter  and  the  tangent  at  its  end  has  exactly 
the  same  form  as  when  the  parabola  is 
referred  to  its  axis  and  the  tangent  at 
the  vertex  (§  108)  it  can  easily  be  shown 
that  the  area  of  any  parabolic  segment  is 
S/3  of  the  area  of  the  circumscribed  paral- 
lelogram formed  by  the  chord,  the  parallel 
tangent,  and  the  two  parallels  to  the  axis 
through  the  extremities  of  the  chord 
(Fig.  67). 


^    JjE  Qi    Ax  Q, 
FlQ.  67 


VI,  §  113]  THE  PARABOLA  115 

With  the  aid  of  this  proposition  Simpson's  rule  can  be  proved  very 
simply.  For,  the  area  of  the  parabolic  segment  P1P3P2  (Fig.  67)  is  then 
equal  to  2/3  of  the  parallelogram  formed  by  the  chord  P1P2,  the  tangent 
at  P2,  and  the  ordinates  yi,  ys  (produced  if  necessary).  This  parallelo- 
gram has  a  height  =  2  Ax  and  a  base  =  ilfP2  =  2/2  —  i  (yi  +  Vs)  ',  hence 
the  area  of  P1P3P2  is  f  Ax  (2  y^^y^-  y^)  =  1  Ax  [4  2/2  -  2  (2/1  +  ys)]. 

To  find  the  whole  shaded  area  we  have  only  to  add  to  this  the  area  of 
the  trapezoid  QiQzPzP\^  which  is  Ax  (1/1+2/3). 

Hence  A  =  QiQsPsP2Pi  =  ^  Ax[4  2/2  -  2(2/1  +  2/3)  +  S(yi  +  2/3)] 
=  ^  Ax(2/i  +42/2  +  2/3). 

EXERCISES 

1.  Show  that  the  area  of  any  parabolic  segment  is  2/3  of  the  area 
of  the  circumscribed  parallelogram. 

2.  In  what  ratio  does  the  parabola  y^  =  4ax  divide  the  area  of  the 
circle  (x  -  a)2  +  2/2  =  4  a^  ? 

3.  Find  the  area  bounded  by  the  parabola  y'^  =  4  ax  and  a  line  of 
slope  m  through  the  focus. 

4.  Find  and  sketch  the  curve  whose  ordinates  represent  the  area 
bounded  by  :  (a)  the  line  ^  =  |  x,  the  axis  Ox,  and  any  ordinate,  (&)  the 
parabola  2/  =  f  x^,  the  axis  Ox,  and  any  ordinate. 

5.  Find  an  approximation  to  the  areas  bounded  by  the  following 
curves  and  the  axis  Ox  (divide  the  interval  in  each  case  into  eight  or 
more  equal  parts) : 

(a)  4  2/  =  16  -  x2.  (6)  2/  =  (x  +  3)  (x  -  2)2.  (c)  2/  =  a;2  -  x^. 

6.  The  cross-sections  in  square  feet  of  a  log  at  intervals  of  6  ft.  are 
3.25,  4.27,  5.34,  6.02,  6.83  ;  find  the  volume. 

7.  The  cross-sections  of  a  vessel  in  square  feet  measured  at  intervals 
of  3  ft.  are  0,  2250,  5800,  8000,  10200  ;  find  the  volume.  Allowing  one 
ton  for  each  35  cu.  ft. ,  what  is  the  displacement  of  the  vessel  ? 

8.  The  half-widths  in  feet  of  a  launch's  deck  at  intervals  of  6  ft.  are 
0,  1.8,  2.6,  3.2,  3.3,  3.3,  2.7,  2.1,  1  ;  find  the  area. 


CHAPTER   VII 


ELLIPSE  AND  HYPERBOLA 


At  Ft 


:^>^ 


fiA 


FiQ.  68 


114.  Definition  of  the  Ellipse.  The  ellipse  may  be  defined 
as  the  locus  of  a  point  whose  distances  from  two  fixed  points  have 
a  constant  sum. 

If  Fi ,  F2  (Fig.  68)  are  the  fixed  points,  which  are  called  the 
focif  and  if  P  is  any  point  of  the 
ellipse,  the  condition  to  be  satisfied 
by  P  is 

FiP  -h  F,P  =  2  a. 

The  ellipse  can  be  traced  mechan- 
ically by  attaching  at  F^y  Ff  the 
ends  of  a  string  of  length  2  a  and 
keeping  the  string  taut  by  means  of  a  pencil.  It  is  obvious 
that  the  curve  will  be  symmetric  with  respect  to  the  line  FiF^, 
and  also  with  respect  to  the  perpendicular  bisector  of  F1F2. 
These  axes  of  symmetry  are  called  the  axes  of  the  ellipse ;  their 
intersection  O  is  called  the  center  of  the  ellipse. 

115.  Axes.  The  points  A^,  A^y  B^,  B^  (Figs.  68  and  69) 
where  the  ellipse  intersects  these  axes  are  called  vertices. 
The  distance  A^A^  of  those  vertices 
that  lie  on  the  axis  containing  the 
foci  -Fi,  i^2  is  =  2  a,  the  length  of 
the  string.  For  when  the  point  P 
in  describing  the  ellipse  arrives  at 
-4i,  the  string  is  doubled  along 
F^A^  so  that 

116 


FiQ.  69 


VII,  §  116]  ELLIPSE  AND  HYPERBOLA  117 

and  since,  by  symmetry,  A2F2  =  i^i A ,  we  have 
A2F2  +  F^F^  +  F^A^  =  A^A^  =  2  a. 
The  distance  AoA^  =  2  a,  which  is  called  the  major  axiSj  must 
evidently  be  not  less  than  the  distance  1^2-^1  between  the  foci, 
which  we  shall  denote  by  2  c. 

The  distance  B2B1  of  the  other  two  vertices  is  called  the 
minor  axis  and  will  be  denoted  by  2  b.     We  then  have 
62  =  a2  -  c2 ; 

for  when  P  arrives  at  Bi ,  we  have  B^F^  =  B^F^  =  a. 

116.  Equation  of  the  Ellipse.  If  we  take  the  center  0  as 
origin  and  the  axis  containing  the  foci  as  axis  Ox,  the  equation 
of  the  ellipse  is  readily  found  from  the  condition  F^P  +  F^P 
=  2  a,  which  gives,  since  the  coordinates  of  the  foci  are  c,  0 
and  —  c,  0 : 

■Vix  -  c)2  +  2/2  +  V(aj  +  c)2  +  2/2  =  2  a. 
Squaring  both  members  we  have 


a.2  _^  2/2  -f-  c2  +  V(a;2  +  2/^  +  02  —  2  cx){x^  +  y^  -^  c'^  -{-  2  ex)  =  2  a^  -, 
transferring  a;2  -f  2/2  +  c2  to  the  right-hand  member  and  squar- 
ing again,  we  find 

(a;2  +  2/2  -f  c2)2  -  4  c2a;2  =  4  a^  -  4  a2(a;2  -}-  2/2  -f-  c^)  +  (x^  +  y^-\-  c^y, 
i.e.  (a2  -  c2)a;2  +  aY  =  a^(a^  -  c2). 

Now  for  the  ellipse  (§  115)  a2  —  c2  =  b^.  Hence,  dividing  both 
members  by  a^b%  we  find 

as  the  cartesian  equation  of  the  ellipse  referred  to  its  axes. 

This  equation  shows  at  a  glance :  (a)  that  the  curve  is  sym- 
metric to  Ox  as  well  as  to  Oy ;  (6)  that  the  intercepts  on  the 
axes  Ox,  Oy  are  ±  a,  and  ±  b.  The  lengths  a,  b  are  called 
the  semi-axes. 


118  PLANE  ANALYTIC  GEOMETRY        [VII,  §  116 

Solving  tlie  equation  for  y  we  find 

(2)  2^  =  ±-V^^'=^, 

ct 

which  shows  that  the  curve  does  not  extend  beyond  the  vertex 

Ai  on  the  right,  nor  beyond  A2  on  the  left. 

If  a  and  h  (or,  what  amounts  to  the  same,  a  and  c)  are  given 

numerically,  we  can  calculate  from  (2)  the  ordinates  of   as 

many  points  as  we  please.     If,  in  particular,  a  =  b  (and  hence 

c  =  0),  the  ellipse  reduces  to  a  circle, 

EXERCISES 

1.  Sketch  the  eUipse  of  semi-axes  a  =  4,  6  =  3,  by  marking  the  ver- 
tices, constructing  the  foci,  and  determining  a  few  points  of  the  curve 
from  the  property  FiP  -\-  F2P=  2  a.  Write  down  the  equation  of  this 
ellipse,  referred  to  its  axes. 

2.  Sketch  the  ellipse  x^/l6  +  y^/9  =  1  by  drawing  the  circumscribed 
rectangle  and  finding  some  points  from  the  equation  solved  for  y. 

3.  Sketch  the  ellipses  :  (a)  ««  +2  y*  =  1.     (6)  3  x^  +  12  y*  =  6. 

(c)  3  a;2  +  3 1/2  =  20.         (d)  x^  +  20  y^  =  1. 

4.  If  in  equation  (1)  a  <  6,  the  equation  represents  an  ellipse  whose 
foci  lie  on  Oy.     Sketch  the  ellipses  : 

(a)   T  +  7^=l'  (^)  20x2  +  2/2  =  1.  (c)  10x2  +  9y2=io. 

4       lo 

6.  Find  the  equation  of  the  ellipse  referred  to  its  axes  when  the  foci 
are  midpoints  between  the  center  and  vertices. 

6.  Find  the  product  of  the  slopes  of  chords  joining  any  point  of  an 
ellipse  to  the  ends  of  the  major  axis.  What  value  does  this  product 
assume  when  the  ellipse  becomes  a  circle  ? 

7.  Derive  the  equation  of  the  ellipse  with  foci  at  (0,  c),  (0,  —  c),  and 
major  axis  2  a. 

8.  Write  the  equations  of  the  following  ellipses  :  (a)  with  vertices 
at  (5,  0),  (-  5,  0),  (0,  4),  (0,  -  4)  ;  (&)  with  foci  at  (2,  0),  (-  2,  0), 
and  major  axis  6. 

9.  Find  the  equation  of  the  ellipse  with  foci  at  (1,  1),  (—1,  —  1), 
and  major  axis  6,  and  sketch  the  curve. 


VII,  §  118]  ELLIPSE  AND  HYPERBOLA 


119 


117.   Definition  of  the  H3rperbola.     The  hyperbola  can  be 

defined  as  the  locus  of  a  point  whose  distances  from  two  fixed 

points  have  a  constant  difference. 

The  fixed  points  F^,  F^  are  again  called  the  foci;  if  2  a  is 

the  constant  difference,  every  point  P  of  the  hyperbola  must 

satisfy  the  condition 

F^P-F^P=^±2a. 

Kotice  that  the  length  2  a  must  here  be  not  greater  than  the 
distance  F^^  =  2  c  of  the  foci. 

The  curve  is  symmetric  to  the  line  -Fa^i  and  to  its  perpen- 
dicular bisector. 

A  mechanism  for  tracing  an  arc  of  a  hyperbola  consists  of 
a  straightedge  F2Q  (Fig.  70)  which  turns  about  one  of  the 
foci,  F2 ;  a  string,   of  length  F2Q  —  2a,  is  fastened  to  the 


Fig.  70 

straightedge  at  Q  and  with  its  other  end  to  the  other  focus, 
Fi.  As  the  straightedge  turns  about  F^,  the  string  is  kept 
taut  by  means  of  a  pencil  at  P  which  describes  the  hyperbolic 
arc.  Of  course  only  a  portion  of  the  hyperbola  can  be  traced 
in  this  manner. 

118.  Equation  of  the  Hyperbola.  If  the  line  F^F^  be  taken 
as  the  axis  Ox,  its  perpendicular  bisector  as  the  axis  Oy,  and  if 
F^F^  =  2  c,  the  condition  F^P-F^P^  ±2  a  becomes  (Fig.  71)  ; 


■y/(x  +  cy  -{-f  -Vix  -  cy  -\-y'  =  ±2  a. 


120  PLANE  ANALYTIC  GEOMETRY 

Squaring  both  members  we  find 


[VII,  §  118 


a^+2/' 4- c^- V(ar^ 4-2^' 4- c^- 2  ca;)(a^H- 2/2 -fc^-f  2  caj)  =2  a^ 
squaring  again  and  reducing  as  in  §  116,  we  find  exactly  the 
same  equation  as  in  §  116 : 


Fig.  71 
But  in  the  present  case  c^  a,  while  for  the  ellipse  we  had 
c  <  a.     We  put,  therefore,  for  the  hyperbola 
c2-a2  =  &2; 

the  equation  then  reduces  to  the  form 


(3) 


a2      62 


which  is  the  cartesian  equation  of  the  hyperbola  referred  to  its  axes. 

119.  Properties  of  the  Hjrperbola.  The  equation  (3)  shows 
at  once:  (a)  that  the  curve  is  symmetric  to  Ox  and  to  Oy\ 
(6)  that  the  intercepts  on  the  axis  Ox  are  ±  a,  and  that  the 
curve  does  not  intersect  the  axis  Oy. 

The  line  F^F^  joining  the  foci  and  the  perpendicular  bisector 
of  F^F^  are  called  the  axes  of  the  hyperbola ;  the  intersection 
0  of  these  axes  of  symmetry  is  called  the  center. 

The  hyperbola  has  only  two  vertices,  viz.  the  intersections 
Ax ,  Ai  with  the  axis  containing  the  foci. 


VII,  §  120]  ELLIPSE  AND  HYPERBOLA  121 

The  shape  of  the  hyperbola  is  quite  different  from  that  of 
the  ellipse.     Solving  the  equation  for  y  we  have 


h 


(4)  y=±-^o?-a?, 

which  shows  that  the  curve  extends  to  infinity  from  A^  to  the 
right  and  from  A^  to  the  left,  but  has  no  real  points  between 
the  lines  x  =  a,  x  =  —  a. 

The  line  F^F^  containing  the  foci  is  called  the  transverse 
axis;  the  perpendicular  bisector  of  F^F^  is  called  the  conjugate 
axis.  The  lengths  a,  b  are  called  the  transverse  and  conjugate 
semi-axes. 

In  the  particular  case  when  a=b,  the  equation  (3)  reduces  to 
7^  —  y^  —  a^f 
and  such  a  hyperbola  is  called  rectangular  or  equilateral. 

120.  As3rmptotes.  In  sketching  the  hyperbola  (3)  or  (4)  it 
is  best  to  draw  first  of  all  the  two  straight  lines 

i.e. 

(5)  y^±U 

a 

which  are  called  the  asymptotes  of  the  hyperbola. 

Comparing  with  equation  (4)  it  appears  that,  for  any  value 
of  X,  the  ordinates  of  the  hyperbola  (4)  are  always  (in  absolute 
value)  less  than  those  of  the  lines  (5);  but  the  difference 
becomes  less  as  x  increases,  approaching  zero  as  x  increases 
indefinitely. 

Thus,  the  hyperbola  approaches  its  asymptotes  more  and 
more  closely,  the  farther  we  recede  from  the  center  on  either 
side,  without  ever  reaching  these  lines  at  any  finite  distance 
from  the  center. 


122  PLANE  ANALYTIC  GEOMETRY      [VII,  §  120 

EXERCISES 

1.  Sketch  the  hyperbola  a;2/16  —  yV4  =  1,  after  drawing  the  asymp- 
totes, by  determiuing  a  few  points  from  the  equation  solved  for  y ;  mark 
the  foci. 

2.  Sketch  the  rectangular  hyperbola  x^  —  y"^  —  9.  Why  the  name 
rectangular  ? 

3.  With  respect  to  the  same  axes  draw  the  hyperbolas : 

(a)  20a;2  -  ys  =  12.  (6)  x^  -  20  y^  =  12.  (c)  x^-y'^z=  12. 

4.  The  equation  —  a^^/a^  +  y'^jlP-  —  1  represents  a  hyperbola  whose 
foci  lie  on  the  axis  Oy.    Sketch  the  curves : 

(a)  -  3x2  +  4j/2  =  24.    (6)  a;2_  3^2  -|-  18  =  0.     (c)  x^-y'^  +  16  =  0. 
6.   Sketch  to  the  same  axes  the  hyperbolas : 

9       ^  '     9       ^ 

Two  such  hyperbolas  having  the  same  asymptotes,  but  with  their  axes 
interchanged,  are  called  conjugate. 

6.  What  happens  to  the  hyperbola  x^/a'^  —  y'^/h'^  =  1  as  a  varies  ?  as 
6  varies  ? 

7.  The  equation  x^/aP'  —  y^/b^  =  k  represents  a  family  of  similar 
hyperbolas  in  which  k  is  the  parameter.  What  happens  as  k  changes 
from  1  to  —  1  ?     What  members  of  this  family  are  conjugate  ? 

8.  Find  the  foci  of  the  hyperbolas : 

(a)  9  a;2  -  16 1/2  =  144.  (6)  3  a;2  _  y2  =  12. 

9.  Find  the  hyperbola  with  foci  (0,  3),  (0,  —  3)  and  transverae  axis  4. 

10,  Find  the  equation  of  the  hyperbola  referred  to  its  axes  when  the 
distance  between  the  vertices  is  one  half  the  distance  between  the  foci. 

11.  Find  the  distance  from  an  asymptote  to  a  focus  of  a  hyperbola. 
13.   Show  that  the  product  of  the  distances  from  any  point  of  a  hyper- 
bola to  its  asymptotes  is  constant. 

13.  Find  the  hyperbola  through  the  point  (1,  1)  with  asymptotes 

y=±2x. 

14.  Find  the  equation  of  the  hyperbola  whose  foci  are  (1,  1), 
(—1,  —  1),  and  transverse  axis  2,  and  sketch  the  curve. 


VII,  §  122]  ELLIPSE  AND  HYPERBOLA  123 

121.  Ellipse  as  Projection  of  Circle.  If  a  circle  be  turned 
about  a  diameter  ^2^1  =  ^  a  through  an  angle  €(<|-7r)  and 
then  projected  on  the  original  plane,  the  projection  is  an 
ellipse. 

For,  if  in  the  original  plane  we  take  the  center  0  as  origin 
and  OAi  as  axis  Ox  (Fig.  72),  the 
ordinate  QP  of  every  point  P  of 
the  projection  is  the  projection  of 
the  corresponding  ordinate  QP^  of 
the  circle;  i.e.  —j^ q 

QP=QPi  cose.  Fig.  72 

The  equation  of  the  projection  is  therefore  obtained  from  the 
equation 

x'  +  y^  =  a' 

of  the  circle  by  replacing  y  by  y/cos  c.     The  resulting  equation 


x^-\- 


y' 


COS^e 

represents  an  ellipse  whose  semi-axes  are  a,  the  radius  of  the 
circle,  and  6  =  a  cos  c,  the  projection  of  this  radius. 

122.   Construction  of  Ellipse  from  Circle.    We  have  just 
seen  that,  if  a  >  6,  the  ellipse 

can  be  obtained  from  its  circumscribed  circle  a^  +  2/^  =  a^  by  re- 
ducing all  the  ordinates  of  this  circle  in  the  ratio  b/a.  This 
also  appears  by  comparing  the  ordinates 


y  =  ±-  V  a^  —  x"^ 


a 


of  the  ellipse  with  the  ordinates  ?/  =  ±  Va^  —  x^  of  the  circle. 


124 


PLANE  ANALYTIC  GEOMETRY       [VII,  §  122 


But  the  same  ellipse  can  also  be  obtained  from  its  inscribed 
circle  a^ -{- y^  =  b^  hj  increasing  each  abscissa  in  the  ratio  a/b, 
as  appears  at  once  by  solving  for  x. 

It  follows  that  when  the  semi-axes  a,  b  are  given,  points  of 
the  ellipse  can  be  constructed  by  drawing  concentric  circles  of 
radii  a,  b  and  a  pair  of  perpendicular  diameters  (Fig.  73) ;  if 


Fig.  73 
any  radius  meets  the  circles  at  Pj ,  P^  >   the  intersection  P  of 
the  parallels  through  P^ ,  Pj  ^o  the  diameters  is  a  point  of  the 

ellipse. 

123.  Tangent  to  Ellipse.  It  follows  from  §  121  that  if 
P  («,  y)  is  any  point  of  the  ellipse  and  Pj  that  point  of  the  cir- 
cumscribed circle  which  has  the  same  abscissa,  the  tangents  at 
P  to  the  ellipse  and  at  P^  to  the  circle  must  meet  at  a  point  T  on 
the  major  axis  (Fig.  74). 


Q     A, 
Fig.  74 
For,  as  the  circle  is  turned  about  A^A^  into  the  position  in 
which  P  is  the  projection  of  Pj ,  the  tangent  to  the  circle  at  Pi 
is  turned  into  the  position  whose  projection  is  PT,  the  point  T 
on  the  axis  remaining  fixed. 


VII,  §  124] 


ELLIPSE  AND  HYPERBOLA 


125 


The  tangent  a^jX  +  2/1  Y=  a^  to  the  circle  at  Pi  {xi ,  y^)  meets 
the  axis  Ox  at  the  point  T  whose  abscissa  is 

OT=ayxi  =  a''/x, 
Hence  the  equation  of  the  tangent  at  P(a;,  y)  to  the  ellipse  is 


IT        «^ 

=  — ,  or 

a2       y 


X  — 


yX-fx-^^Y-a^^=0: 


X 


dividing  by  a^y/x  and  observing  that,  by  the  equation  of  the 
ellipse,  x^  —  o?  =  —  {a?/lf)y'^  we  find 


(6) 


xX     yY. 


as  equation  of  the  tangent  to  the  ellipse  (1)  at  4lie  point  P(x,  y). 

It  follows  from  the  equation  of  the  tangent  that  the  slope 
of  the  ellipse  at  any  point  P(x,  y)  is 

¥x 


tan  a  = 


a^y 


124.  Eccentricity.  For  the  length  of  the  focal  radius  F^P 
of  any  point  P{x,  y)  of  the  ellipse  (1)  we  have  (Fig.  75), 
since  a^  —  6^  =  c^ : 

F,P^={x-cy+y^=(x-cy-{-—(a''-x^)=^^(a'-  2  a''cx-{-c'x% 


whence       FiP=± 


{-i^y 


The  ratio  c/a  of  the  distance 
2  c  of  the  foci  to  the  major  axis 
2  a  is  called  the  (numerical)  ec- 
centricity of  the  ellipse.  Denot- 
ing it  by  e  we  have  F^P=  ±  (a  —  ex), 
and  similarly  we  find  F^P^  ±  (a  +  ex). 


Fig.  75 


126 


PLANE  ANALYTIC  GEOMETRY      [VII,  §  124 


For  the  hyperbola  (3)  we  find  in  the  same  way,  if  we  again 
put  e  =  cla,  exactly  the  same  expressions  for  the  focal  radii 
F^P^  F^P (in  absolute  value).  But  as  for  the  ellipse  c^^o?—  b^ 
while  for  the  hyperbola  c^^a^  +  b-  it  follows  that  the  eccentric- 
ity of  the  ellipse  is  always  a  proper  fraction  becoming  zero  only 
for  a  circle^  while  the  eccentricity  of  the  hyperbola  is  always  greater 
than  one. 

125.  Equation  of  Normal  to  Ellipse.  As  the  normal  to  a 
curve  is  the  perpendicular  to  its  tangent  through  the  point  of 
contact,  the  equation  of  the  normal  to  the  ellipse  (1)  at  the  point 
P{Xy  y)  is  readily  found  from  the  equation  (6)  of  the  tangent  as 

y  X  ^         /I       IN        c' 


I.e. 


^X--Y=c\ 
X  y 


The  intercept  made  by  this  normal  on  the  axis  Ox  is  there- 
fore 

ON=z^x=^e'hi. 


From  this  result  it  appears  by  §  125  that  (Fig.  76) 

FiN=  c-\-e^x=i  e(a  -f-ea;)  =  e  •  F^P, 
FiN=  c-e^  =  e(a-ex)=e'  F^P; 

hence  the  normal  divides  the  dis- 
tance F2F1  in  the  ratio  of  the 
adjacent  sides  F^P^  F^P  of  the 
triangle  F^PF.^.  It  follows  that 
the  normal  bisects  the  angle  betiveen 
the  focal  radii  PF^ ,  PF^ ;  in  other  words,  the  focal  radii  are 
equally  inclined  to  the  tangent. 


Fia.  76 


VII,  §  127]  ELLIPSE  AND  HYPERBOLA  127 

126.   Construction  of   any  Hyperbola   from  Rectangular 
Hyperbola.     The  ordinates  (4), 


y  =  ±-  Va;2 


a 


of  the  hyperbola  (3)  are  b/a  times  the  corresponding  ordinates 


y  =  ±  'vx^  —  a^ 

of  the  equilateral  hyperbola  (end  of  §  119)  having  the  same 
transverse  axis.  When  6  <  a,  we  can  put  b/a  =  cos  e  and  re- 
gard the  general  hyperbola  as  the  projection  of  the  equilateral 
hyperbola  of  equal  transverse  axis.  When  6  >  a,  we  can  put 
a/b  =  cos  €  so  that  the  equilateral  hyperbola  can  be  regarded  as 
the  projection  of  the  general  hyperbola. 

In  either  case  it  is  clear  that  the  tangents  to  the  general  and 
equilateral  hyperbolas  at  corresponding  points  (i.e.  at  points 
having  the  same  abscissa)  must  intersect  on  the  axis  Ox. 

127.  Slope  of  Equilateral  Hyperbola.  To  find  the  slope  of 
the  equilateral  hyperbola 

X'^  —  y^  —  a^j 

observe  that  the  slope  of  any  secant  joining  the  point  P{x,  y) 
and  Pi{xyy  2/i)  ^^  (2/i~2/)/(^i~^)>  ^-nd  that  the  relations 

2/2=aj2-a2,  y^  =  x^—o? 
give      y^-yi^  =  x^-x^\  i.e.  (y-yi)(y +  yi)  =(x-^x{){x-\'x;)f 

whence  .VZL^ _ ^"^    i . 

x-x^     y+yi 

Hence,  in  the  limit  when  Pi  comes  to  coincidence  with  P,  we 
find  for  the  slope  of  the  tangent  at  P(x,  y)  :  tan  a  =x/y.  Hence 
the  equation  of  the  tangent  to  the  equilateral  hyperbola  is 

T-y  =  -{X-x),oTxX-yT=a\ 

y 

since  x^  —y^^  a^. 


128  PLANE  ANALYTIC  GEOMETRY       [VII,  §  128 

128.  Tangent  to  the  Hyperbola.  It  follows  as  in  §  123  that 
the  tangent  to  the  general  hyperbola  (3)  has  the  equation 

(7)  ^-yl=i, 

^  ^  a^       ¥ 

The  slope  of  the  hyperbola  (3)  is  therefore 

tan  a  = 

a^y 

Notice  that  the  equations  (6),  (7)  of  the  tangents  are  obtained 
from  the  equations  (1),  (3)  of  the  curves  by  replacing  x*,  y'^  by 
xXj  yY,  respectively  (compare  §§  54,  98). 

It  is  readily  shown  (compare  §  126)  that  for  the  hyperbola 
(3)  the  tangent  meets  the  axis  Ox  at  the  point  T  that  divides 
the  distance  of  the  foci  F^Fi  proportionally  to  the  focal  radii 
F2P,  FiP,  so  that  the  tangent  to  the  hyperbola  bisects  the  angle 
between  the  focal  radii. 

EXERCISES 

1.  Show  that  a  right  cylinder  whose  cross-section  (i.e.  section  at 
right  angles  to  the  generators)  is  an  ellipse  of  semi-axes  a,  b  has  two 
(oblique)  circular  sections  of  radius  a ;  find  their  inclinations  to  the 
cross-section. 

2.  Derive  the  equation  of  the  normal  to  the  hyperbola  (3). 

3.  Find  the  polar  equations  of  the  ellipse  and  hyperbola,  with  the 
center  as  pole  and  the  major  (transverse)  axis  as  polar  axis. 

4.  Find  the  lengths  of  the  tangent,  subtangent,  normal,  and  sub- 
normal in  terms  of  the  coordinates  at  any  point  of  the  ellipse. 

6,  Show  that  an  ellipse  and  hyperbola  with  common  foci  are 
orthogonal. 

6.  Show  that  the  eccentricity  of  a  hyperbola  is  equal  to  the  secant 
of  half  the  angle  between  the  asymptotes. 

7.  Express  the  cosine  of  the  angle  between  the  asymptotes  of  a 
hyperbola  in  terms  of  its  eccentricity. 

8.  Show  that  the  tangents  at  the  vertices  of  a  hyperbola  intersect  the 
asymptotes  at  points  on  the  circle  about  the  center  through  the  foci. 


VII,  §  128]  ELLIPSE  AND   HYPERBOLA  129 

9.  Show  that  the  point  of  contact  of  a  tangent  to  a  hyperbola  is  the 
midpoint  between  its  intersections  with  the  asymptotes. 

10.  Show  that  the  area  of  the  triangle  formed  by  the  asymptotes  and 
any  tangent  to  a  hyperbola  is  constant. 

11.  Show  that  the  product  of  the  distances  from  the  center  of  a  hyper- 
bola to  the  intersections  of  any  tangent  with  the  asymptotes  is  constant. 

12.  Show  that  the  tangent  to  a  hyperbola  at  any  point  bisects  the  angle 
between  the  focal  radii  of  the  point. 

13.  As  the  sum  of  the  focal  radii  of  every  point  of  an  ellipse  is  con- 
stant (§  116)  and  the  normal  bisects  the  angle  between  the  focal  radii 
(§  125),  a  sound  wave  issuing  from  one  focus  is  reflected  by  the  ellipse 
to  the  other  focus.  This  is  the  explanation  of  "whispering  galleries." 
Find  the  semi-axes  of  an  elliptic  gallery  in  which  sound  is  reflected  from 
one  focus  to  the  other  at  a  distance  of  69  ft.  in  1/10  sec.  (the  velocity  of 
sound  is  1090  ft, /sec). 

14.  Show  that  the  distance  from  any  point  of  an  equilateral  hyperbola 
to  its  center  is  a  mean  proportional  to  the  focal  radii  of  the  point. 

15.  Show  that  the  bisector  of  the  angle  formed  by  joining  any  point 
of  an  equilateral  hyperbola  to  its  vertices  is  parallel  to  an  asymptote. 

16.  Show  that  the  tangents  at  the  extremities  of  any  diameter  (chord 
through  the  center)  of  an  ellipse  or  hyperbola  are  parallel. 

17.  Let  the  normal  at  any  point  Pof  an  ellipse  referred  to  its  axes  cut 
the  coordinate  axes  at  Q  and  B  ;  find  the  ratio  PQ/PB. 

18.  Show  that  a  tangent  at  any  point  of  the  circle  circumscribed  about 
an  ellipse  is  also  a  tangent  to  the  circle  with  center  at  a  focus  and  radius 
equal  to  the  focal  radius  of  the  corresponding  point  of  the  ellipse. 

19.  Show  that  the  product  of  the  y-intercept  of  the  tangent  at  any 
point  of  an  ellipse  and  the  ordinate  of  the  point  of  contact  is  constant. 

20.  Find  the  locus  of  the  center  of  a  circle  which  touches  two  fixed 
non-intersecting  circles. 

21.  Find  the  locus  of  a  point  at  which  two  sounds  emitted  at  an 
interval  of  one  second  at  two  points  2000  ft.  apart  are  heard  simul- 
taneously. 


130  PLANE  ANALYTIC  GEOMETRY        [VII,  §  129 

129.  Intersections   of  a   Straight   Line  and  an  Ellipse. 

The  intersections  of  the  ellipse  (1)  with  any  straight  line  are 
found  by  solving  the  simultaneous  equations 

y  =  mx  4-  k. 
Eliminating  y,  we  find  a  quadratic  equation  in  x: 

(m'a}  +  62)a^  ^  2  mka'x  +  {Jc"  -  6')a'  =  0. 
To  each  of  the  two  roots  the  corresponding  value  of  y  results 
from  the  equation  y  =  mx  +  k. 

Thus,  a  straight  line  can  intersect  an  ellipse  in  not  more  than 
two  points. 

130.  Slope  Form  of  Tangent  Equations.  If  the  roots  of 
the  quadratic  equation  are  equal,  the  line  has  but  one  point  in 
common  with  the  ellipse  and  is  a  tangent. 

The  condition  for  equal  roots  is 

m'lea^  =  {m^a^  +  &')(*:*  -  &*), 
whence  k=±  Vm^a^  +  b\ 

The  two  parallel  lines 
(8)  y  =  mx ±  VmV+^ 

are  therefore  tangents  to  the  ellipse  (1),  whatever  the  value  of 
m.  This  equation  is  called  the  slope  form  of  the  equation  of  a 
tangent  to  the  ellipse. 

It  can  be  shown  in  the  same  way  that  a  straight  line  cannot 
intersect  a  hyperbola  in  more  than  two  points,  and  that  the 
two  parallel  lines 

y  =  WM5±  -Vm^a^  —  6' 

have  each  but  one  point  in  common  with  the  hyperbola  (3). 

131.  The  condition  that  a  line  be  a  tangent  to  an  ellipse  or 
hyperbola  assumes  a  simple  form  also  when  the  line  is  given 
in  the  general  form 

Ax-{-By+C=0. 


VII,  §  132]  ELLIPSE  AND  HYPERBOLA  131 

Substituting  the  value  of  y  obtained  from  this  equation  in 
the  equation  (1)  of  the  ellipse,  we  find  for  the  abscissas  of  the 
points  of  intersection  the  quadratic  equation : 

{A^a?  +  jB262)ic2  +  2  ACo^x  +  (O^  -  I^h'')a^  =  0; 
the  condition  for  equal  roots  is 

A^C^a^  =  {AH^  +  Wh^C^  -  B'b''), 

which  reduces  to 

A^a'-^-B'b'^C'. 

The  line  is  therefore  a  tangent  whenever  this  condition  is 
satisfied. 

When  the  line  is  given  in  the  normal  form, 

a;  cos  )8  +  2/  sin  ^  =  j9, 

the  condition  becomes 

132.  Tangents  from  an  Exterior  Point.    By  §  130  the  line 


y  =  mx  +  V  m'-^a'-^  +  b^^ 

is  tangent  to  the  ellipse  (1)  whatever  the  value  of  m.     The  condition  that 
this  Une  pass  through  any  given  point  {xi ,  yi)  is 


2/1  =  mxi  +  y/m^d^  +  6^  ; 
transposing  the  term  mxi,  and  squaring,  we  find  the  following  quadratic 

equation  for  m  : 

mHx^  -  2  mxxyx  +  y^  =  m'^a^  +  b^, 
i.e.  (xi^  -  a^)m^  -  2  xiyim  +  yi^  -b'^  =  0. 

The  roots  of  this  equation  are  the  slopes  of  those  lines  through  (xi ,  yi) 
that  are  tangent  to  the  ellipse  (1). 

Thus,  not  more  than  two  tangents  can  be  drawn  to  an  ellipse  from  any 
point.  Moreover,  these  tangents  are  real  and  different,  real  and  coin- 
cident, or  imaginary,  according  as 


132  PLANE  ANALYTIC  GEOMETRY      [VII,  §  132 

This  condition  can  also  be  written  in  the  form 

i.e.  ^\yl^i^o. 

a^      h"^        < 

Hence,  to  see  whether  real  tangents  can  be  drawn  from  a  point  (xi ,  yi) 
to  the  ellipse  (1)  we  have  only  to  substitute  the  coordinates  of  the  point 
for  X,  y  in  the  expression 

^  +  2^-1- 

if  the  expression  is  zero,  the  point  (xi,  yi)  lies  on  the  ellipse,  and  only 
one  tangent  is  possible ;  if  the  expression  is  positive,  two  real  tangents 
can  be  drawn,  and  the  point  is  said  to  he  outside  the  eUipse  ;  if  the  expres- 
sion is  negative,  no  real  tangents  exist,  and  the  point  is  said  to  he  within 
the  eUipse. 

These  definitions  of  inside  and  outside  agree  with  what  we  would 
naturally  call  the  inside  or  outside  of  the  eUipse.  But  the  whole  discus- 
sion applies  equally  to  the  hyperbola  (3)  where  the  distinction  between 
inside  and  outside  is  not  so  obvious. 

133.  Sjonmetry.  Since  the  ellipse,  as  well  as  the  hyperbola, 
has  two  rectangular  axes  of  symmetry,  the  axes  of  the  curve, 
it  has  a  center ^  the  intersection  of  these  axes,  i.e.  a  point  of 
symmetry  such  that  every  chord  through  this  point  is  bisected 
at  this  point  (compare  §  70).  Analytically  this  means  that 
since  the  equation  (1),  as  well  as  (3),  is  not  changed  by  replac- 
ing a;  by  —  ic,  nor  by  replacing  yhy  —y,  it  is  not  changed  by 
replacing  both  x  and  yhy  —  x  and  —  y,  respectively.  In  other 
words,  if  (a;,  y)  is  a  point  of  the  curve,  so  is  (—  a;,  —  y).  This 
fact  is  expressed  by  saying  that  the  origin  is  a  point  of  sym- 
metry, or  center. 

134.  Conjugate  Diameters.  Any  chord  through  the  center 
of  an  ellipse  or  hyperbola  is  called  a  diameter  of  the  curve. 


VII,  §  134]  ELLIPSE  AND  HYPERBOLA 


133 


Just  as  in  the  case  of  the  circle,  so  for  the  ellipse  the  locus 
of  the  midpoints  of  any  system  of  parallel  chords  is  a  diameter. 
This  follows  from  the  corresponding  property  of  the  circle 
because  the  ellipse  can  be  regarded  as  the  projection  of  a 
circle  (§121).  But  this  diameter  is  in  general  not  perpen- 
dicular to  the  parallel  chords ;  it  is  said  to  be  conjugate  to  the 
diameter  that  occurs  among  the  parallel  chords.  Thus,  in  Fig. 
77,  P'Q'  is  conjugate  to  PQ  (and  vice  versa). 


Fig.  77 

To  find  the  diameter  conjugate  to  a  given  diameter  y  =  mx 
of  the  ellipse  (1),  let  y—mx  -\-k  be  any  parallel  to  the  given 
diameter.  If  this  parallel  intersects  the  ellipse  (1)  at  the  real 
points  (a^,  2/1)  and  (X2, 2/2)?  the  midpoint  has  the  coordinates 
^(ajj  +  X2),  J(yi  +  2/2)-     The  quadratic  equation  of  §  129  gives 


X=-{Xi-{-X2)  = 


ma^k 


m^a^  +  62 

If  instead  of  eliminating  y  we  eliminate  x,  we  obtain  the  quad- 
ratic equation 

(m2a2+ 62)2/2  _  2  Wy  +  Qc"  -  m'  a')W  =  0, 

whence 


y 


1/      ,      N  ^'^ 

7^  (2/1  +  2/2)  =  —TTTTi 
2  mV+d^ 


Eliminating  k  between  these  results,  we  find  the  equation  of  the 
locus  of  the  midpoints  of  the  parallel  chords  of  slope  m : 


134  PLANE  ANALYTIC  GEOMETRY      [VII,  §  134 

(9)  y  =  --\x, 

^  ^  ma? 

If  m  =  tan  a  is  the  slope  of  any  diameter  of  the  ellipse  (1), 
the  slope  of  the  conjugate  diameter  is 

mi  =  tan  «!  = -• 

The  diameter  conjugate  to  this  diameter  of  slope  wij  has  there- 
fore the  slope 

mo=  — 


''  -     m,a' 


\     ma^J 


i.e.  it  is  the  original  diameter  of  slope  m  (Fig.  77).  In  other 
words,  either  one  of  the  diameters  of  slopes  m  and  m^  is  conjugate 
to  the  other  ;  each  bisects  the  chords  parallel  to  the  other. 

135.  Tangents  Parallel  to  Diameters.  Among  the  parallel 
lines  of  slope  m,  y  =  mx -{- k,  there  are  two  tangents  to  the 
ellipse,  viz.  (§  130)  those  for  which 

k=±VrrM^f¥, 

their  points  of  contact  lie  on  (and  hence  determine)  the  conju- 
gate diameter.  This  is  obvious  geometrically;  it  is  readily 
verified  analytically  by  showing  that  the  coordinates  of  the 
intersections  of  the  diameter  of  slope  —  b^/ma^  with  the 
ellipse  (1)  satisfy  the  equations  of  the  tangents  of  slope  m,  viz. 

y  =  mx  ±  -y/m^a^  +  b^. 

The  tangents  at  the  ends  of  the  diameter  of  slope  m  must  of 
course  be  parallel  to  the  diameter  of  slope  wij.  The  four  tan- 
gents at  the  extremities  of  any  two  conjugate  diameters  thus 
form  a  circumscribed  parallelogram  (Fig.  77). 

The  diameter  conjugate  to  either  axis  of  the  ellipse  is  the 
other  axis  ;  the  parallelogram  in  this  case  becomes  a  rectangle. 


VII,  §  136] 


ELLIPSE  AND  HYPERBOLA 


135 


136.  Diameters  of  a  Hjrperbola.  For  the  hyperbola  the 
same  formulas  can  be  derived  except  that  6^  is  replaced 
throughout  by  —  fe^.  But  the  geometrical  interpretation  is 
somewhat  different  because  a  line  y  =  mx  meets  the  hyperbola 
(3)  in  real  points  only  when  m  <  b/a. 


Fig.  78 
The  solution  of  the  simultaneous  equations 

y  =  mx,     y^x^  —  a^y"^  =  a'^ft* 
gives : 


ah 


y- 


mob 


V6^  —  m^a^  V6^  —  m^a} 

These  values  are  real  if  m  <  h/a  and  imaginary  if  m'^hfa 
(Fig.  78).  In  the  former  case  it  is  evidently  proper  to  call  the 
distance  PQ  between  the  real  points  of  intersection  a  diameter 
of  the  hyperbola  ;  its  length  is 

PQ  ==  2  V^^Tf  =  2ab  Jj  +  ^'^- 

If  m>b/a,  this  quantity  is  imaginary;  but  it  is  customary  to 
speak  even  in  this  case  of  a  diameter,  its  length  being  defined 
as  the  real  quantity 

By  this  convention  the  analogy  between  the  properties  of  the 
ellipse  and  hyperbola  is  preserved. 


136  PLANE  ANALYTIC  GEOMETRY      [VII,  §  137 

137.  Conjugate  Diameters  of  a  Hyperbola.  Two  diameters 
of  the  hyperbola  are  called  conjugate  if  their  slopes  m,  mi  are 
such  that 

mmi  =— • 
a* 

One  of  these  lines  evidently  meets  the  curve  in  jeal  points,  the 
other  does  not. 

If  m  <  6/a,  the  line  y  =  mx,  as  well  as  any  parallel  line, 
meets  the  hyperbola  (3)  in  two  real  points,  and  the  locus  of  the 
midpoints  of  the  chords  parallel  to  ^  =  ma;  is  found  to  be  the 
diameter  conjugate  to  y  =  mx,  viz. 

y  =  77hx= — -X. 

If  m  >  6/a,  the  coordinates  x^^  y^  and  aja,  2/2  of  the  intersec- 
tions of  y  =  mx  with  the  hyperbola  are  imaginary;  but  the 
arithmetic  means  ^  (xi  +  x^),  ^(^1  +  2/2)  ^^^  ^^^^)  a-ud  the  locus 
of  the  points  having  these  coordinates  is  the  real  line 

y  =  rriiX  = X. 

ma} 

It  may  finally  be  noted  that  what  was  in  §  136  defined  as 
the  length  of  a  diameter  that  does  not  meet  the  hyperbola 

in  real  points  is  the  length  of  the  real  diameter  of  the  hyper- 
bola 

a'  +  6«       ' 
two  such  hyperbolas  are  called  conjugate. 


VII,  §  138]  ELLIPSE  AND  HYPERBOLA 


137 


138.  Parameter  Equations.    Eccentric  Angle.    Just  as  the 
parameter  equations  of  the  circle  x"^  -{-  y^  =  a^  are  (§  106) ; 

x  =  a  cos  9,  y  =  a  sin  0, 
so  those  of  the  ellipse  (1)  are 

x=:a  cos  $,  y  =  b  sin  Oj 
and  those  of  the  hyperbola  (3)  are 

x=  a  sec  6,  y  =  b  tan  6. 
In  each  case  the  elimination  of  the  parameter  6  (by  squaring 
and  then  adding  or  subtracting)  leads  to  the  cartesian  equation. 

The  angle  0,  in  the  case  of  the 
circle,  is  simply  the  polar  angle  of 
the  point  P{x,  y).  In  the  case  of  the 
ellipse,  as  appears  from  Fig.  79 
(compare  §  122),  6  is  the  polar  angle 
not  of  the  point  P  (x,  y)  of  the  ellipse, 
but  of  that  point  Pi  of  the  circum- 
scribed circle  which  has  the  same 
abscissa  as  P,  and  also  of  that  point 
Pg  of  the  inscribed  circle  which  has  the  same  ordinate  as  P. 
This  angle  6  =  xOP^  is  called  the  eccentric  angle  of  the  point 
P  (x,  y)  of  the  ellipse. 

In  the  case  of  the  hyperbola  the  eccentric  angle  6  determines 
the  point  P{x,  y)  as  follows  (Fig.  80).  '  Let  a  line  through  0 
inclined  at  the  angle  0  to  the  trans- 
verse axis  meet  the  circle  of  radius 
a  about  the  center  at  A,  and  let  the 
transverse  axis  meet  the  circle  of 
radius  b  about  the  center  at  B.  Let 
the  tangent  at  A  meet  the  transverse 
axis  at  A'  and  the  tangent  at  B  meet 
the  line  OA  at  B'.  Then  the  parallels  to  the  axes  through  A 
and  B'  meet  at  P. 


Fia.  79 


y 

-^N 

a 

^%:-^Y 

X 

■  0 

r  b    b                      A' 

Fia.  80 


138  PLANE  ANALYTIC  GEOMETRY      [VII,  §  139 

139.  Area  of  Ellipse.  Since  any  ellipse  of  semi-axes  a,  b 
can  be  regarded  as  the  projection  of  a  circle  of  radius  a, 
inclined  to  the  plane  of  the  ellipse  at  an  angle  c  such  that 
cos  €  =  6/a,  the  area  A  of  the  ellipse  is  ^  =  -n-a^  cos  e  =  irdb. 

EXERCISES 

1.  Find  the  tangents  to  the  ellipse  x^  +  4  j/^  =  16,  which  pass  through 
the  following  points : 

(a)  (2,  V3),    (6)  (-3,  iV7),    (c)  (4,0),    {d)  (-8,0). 

2.  Find  the  tangents  to  the  hyperbola  2  a:^  —  3  ?/2  =  18,  which  pass 
through  the  following  points  : 

(a)  (-6,  3V2),    (6)  (-3,0),    (c)   (4,  -V5),    {d)  (0,0). 

3.  Find  the  intersections  of  the  line  x  —  2  y  =  7  and  the  hyperbola 

x2  -  ?/2  =  5. 

4.  Find  the  intersections  of  the  line  3x-j-2/  —  1=0  and  the  ellipse 

x2  +  4  y2  =  65. 
6.  For  what  value  of  k  will  the  line  i/  =  2x  +  A;bea  tangent  to  the 
hyperbola  x2-4  2/2_4  =  o? 

6.  For  what  values  of  m  will  the  line  y=  mx  +  2  be  tangent  to  the 
ellipse  x2-f4y2_i=o? 

7.  Find  the  conditions  that  the  following  lines  are  tangent  to  the 
hyperbola  x2/a2  -  ^2/52  _  1 . 

(a)  Ax  -\-  By  +  C  =  (i,    (&)  x  cos  /3  +  «/  sin  /3  =  p. 

8.  Are  the  following  points  on,  outside,  or  inside  the  ellipse  x2  -f  4  y2 = 4p 

(a)  (!,f),    (?>)  a,  -i)»    (c)  (-i-l). 

9.  Are  the  following    points  on,  outside,  or  inside  the  hyperbola 
9x2-y2  =  9?  (a)  (f,  -  f ),    (6)  (1.35,2.15),    (c)  (1.3,2.6). 

10.  Find  the  difference  of  the  eccentric  angles  of  points  at  the  extremi- 
ties of  conjugate  diameters  of  an  ellipse. 

11.  Show  that  conjugate  diameters  of  an  equilateral  hyperbola  are 
equal. 

12.  Show  that  an  asymptote  is  its  own  conjugate  diameter. 

13.  Show  that  the  segments  of  any  line  between  a  hyperbola  and  its 
asymptotes  are  equal. 

14.  Find  the  tangents  to  an  ellipse  referred  to  its  axes  which  have 
equal  intercepts. 


VII,  §  1391  ELLIPSE  AND  HYPERBOLA  139 

15.  What  is  the  greatest  possible  number  of  normals  that  can  be  drawn 
from  a  given  point  to  an  ellipse  or  hyperbola  ? 

16.  Show  that  tangents  drawn  at  the  extremities  of  any  chord  of  an 
ellipse  (or  hyperbola)  intersect  on  the  diameter  conjugate  to  the  chord. 

17.  Show  that  lines  joining  the  extremities  of  the  axes  of  an  ellipse 
are  parallel  to  conjugate  diameters. 

18.  Show  that  chords  drawn  from  any  point  of  an  ellipse  to  the  ex- 
tremities of  a  diameter  are  parallel  to  conjugate  diameters. 

19.  Find  the  product  of  the  perpendiculars  let  fall  to  any  tangent  from 
the  foci  of  an  ellipse  (or  hyperbola). 

20.  The  earth's  orbit  is  an  ellipse  of  eccentricity  .01677  with  the  sun 
at  a  focus.  The  mean  distance  (major  semi-axis)  between  the  sun  and 
earth  is  93  million  miles.  Find  the  distance  from  the  sun  to  the  center 
of  the  orbit. 

21.  Find  the  sum  of  the  squares  of  any  two  conjugate  semi-diametere 
of  an  ellipse.  Find  the  difference  of  the  squares  of  conjugate  semi-diam- 
eters of  a  hyperbola. 

22.  Find  the  area  of  the  parallelogram  circumscribed  about  an  ellipse 
with  sides  parallel  to  any  two  conjugate  diameters. 

23.  Find  the  angle  between  conjugate  diameters  of  an  ellipse  in  terms 
of  the  semi-diameters  and  semi-axes. 

24.  Express  the  area  of  a  triangle  inscribed  in  an  ellipse  referred  to 
its  axes  in  terms  of  the  eccentric  angles  of  the  vertices. 

25.  The  circle  which  is  the  locus  of  the  intersection  of  two  perpendicu- 
lar tangents  to  an  ellipse  or  hyperbola  is  called  the  director-circle  of  the 
conic.     Find  its  equation  :  (a)  For  the  ellipse.     (6)  For  the  hyperbola. 

26.  Find  the  locus  of  a  point  such  that  the  product  of  its  distances 
from  the  asymptotes  of  a  hyperbola  is  constant.  For  what  value  of  this 
constant  is  the  locus  the  hyperbola  itself  ? 

27.  Find  the  locus  of  the  intersection  of  normals  drawn  at  correspond- 
ing points  of  an  ellipse  and  the  circumscribed  circle. 

28.  Two  points  J.,  5  of  a  line  I  whose  distance  is  AB  =  a  move  along 
two  fixed  perpendicular  lines  ;  find  the  path  of  any  point  P  oil. 


CHAPTER  VIII 
CONIC    SECTIONS  — EQUATION   OF   SECOND   DEGREE 

PART  I.     DEFINITION   AND  CLASSIFICATION 

140.  Conic  Sections.  The  ellipse,  hyperbola,  and  parabola 
are  together  called  conic  sections,  or  simply  conies,  because 
the  curve  in  which  a  right  circular  cone  is  intersected  by  any 
plane  (not  passing  through  the  vertex)  is  an  ellipse  or  hyper- 
bola according  as  the  plane  cuts  only  one  of  the  half-cones  or 
both,  and  is  a  parabola  when  the  plane  is  parallel  to  a  gener- 
ator of  the  cone.  This  will  be  proved  and  more  fully  dis- 
cussed in  §§  148-152. 

141.  General  Definition.  The  three  conies  can  also  be 
defined  by  a  common  property  in  the  plane :  the  locus  of  a  point 
for  ivhich  the  ratio  of  its  distances  from  a  fixed  point  and  from 
a  fixed  line  is  constant  is  a  conic,  viz.  an  ellipse  if  the  constant 
ratio  is  less  than  one,  a  hyperbola  if 

the  ratio  is  greater  than  one,  and  a . 

parabola  if  the  ratio  is  equal  to  one. 

We   shall   find   that  this  constant 
ratio  is  equal  to  the  eccentricity  e  =  c/a 
as  defined  in  §  124.     Just  as  in  the 
case   of  the  parabola  for  which   the   — 
above  definition  agrees  with  that  of  Fio.  81 

§  89,  we  shall  call  the  fixed  line  d^  directrix,  and  the  fixed 
point  Fi  focus  (Fig.  81). 

142.  Polar  Equation.  Taking  the  focus  F^  as  pole,  the 
perpendicular  from  Fi  toward  the  directrix  dj  as  polar  axis, 

140 


VIII,  §  143]  CONIC  SECTIONS  141 

and  putting  the  given  distance  FiD  =  q,  we  have  FiP=rj 
PQz=q—r  cos  <!>,  r  and  <f>  being  the  polar  coordinates  of  any 
point  P  of  the  conic.  The  condition  to  be  satisfied  by  the 
point  P,  viz.  F^P/PQ  =  e,  i.e.  F^P—  e  •  PQ,  becomes,  therefore, 

r  =  e(g  — rcos<^),  or  r=  ^ 


1  +  e  cos  <^ 

This  then  is  the  polar  equation  of  a  conic  if  the  focus  is  taken 
as  pole  and  the  perpendicular  from  the  focus  toward  the  directrix 
as  polar  axis.  It  is  assumed  that  q  is  not  zero;  the  ratio  e 
may  be  any  positive  number. 

143.  Plotting  the  Conic.  By  means  of  this  polar  equation 
the  conic  can  be  plotted  by  points  when  e  and  q  are  given. 
Thus,  for  <^  =  0  and  <^  =  tt,  we  find  eq/{l  +  e)  and  eq/{l  —  e)  as 
the  intercepts  FiA^  and  jF\^2  on  the  polar  axis ;  A  >  ^2  are  the 
vertices.  For  any  negative  value  of  </>  (between  0  and  —  tt) 
the  radius  vector  has  the  same  length  as  for  the  same  positive 
value  of  <f>.  The  segment  LL'  cut  off  by  the  conic  on  the  per- 
pendicular to  the  polar  axis  drawn  through  the  pole  is  called 
the  latus  rectum;  its  length  is  2  eg.  Notice  that  in  the  ellipse 
and  hyperbola,  i.e.  when  e  ^1,  the  vertex  Ai  does  not  bisect 
the  distance  FiD  (as  it  does  in  the  parabola),  but  that 
F,A,/A,D  =  e. 

If  in  Fig.  81,  other  things  being 
equal,  the  sense  of  the  polar  axis  be 
reversed,  we  obtain  Fig.  82.  We  have 
again  FiP=r;  but  the  distance  of  P 
from  the  directrix  di  is  QP  =  q  + 
r  cos  <f>,  so  that  the  polar  equation  of 
the  conic  is  now : 

r  =  ,       '1        . 
1  —  e  cos  </) 


<i 

y 

^P 

~L, 

D 

Ail     \ 

di 

Fig.  82 

a; 

142  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  144 

144.  Classification  of  Conies.  For  e  =  l,  the  equations  of 
§§  142-143  reduce  to  the  equations  of  the  parabola  given  in 
§§  89,  90.  It  remains  to  show  that  for  e  <  1  and  e  >  1 
these  equations  represent  respectively  an  ellipse  and  a  hyper- 
bola as  defined  in  §§  114,  117. 

To  show  this  we  need  only  introduce  cartesian  coordi- 
nates and  then  transform  to  the  center,  i.e.  to  the  midpoint 
0  between  the  intersections  Ai,  A2  of  the  curve  with  the  polar 
axis. 

145.  Transformation  to  Cartesian  Coordinates.  The  equa- 
tion of  §  142, 

r  =  e(q  —  r  cos  <^) 

becomes  in  cartesian  coordinates,  with  the  pole  Fi  as  origin 
and  the  polar  axis  as  axis  Ox  (Fig.  81) : 


V?+y  =  e{q  —  x)y 
or,  rationalized : 

(1  -  e2)flj2  ^2e^qx-hf  =  e\^. 

The  midpoint  0  between  the  vertices  A^^  A^  at  which  the 
curve  meets  the  axis  Ox  has,  by  §  143,  the  abscissa 

2   ^Vl  +  e      \-e)          l-e^' 
this  also  follows  from  the  cartesian  equation,  with  ?/  =  0. 

146.  Change  of  Origin  to  Center.  To  transform  to  paral- 
lel axes  through  this  point  O  we  have  to  replace  x  by 
X  —  e^q/(^.  —  e?) ;  the  equation  in  the  new  coordinates  is  there- 
fore 

and  this  reduces  to 

(1  -  e^^  +  /  =  eV(l  +  j^)  =  ^, 


VIII,  §  147]  CONIC  SECTIONS  143 

(1  _  ^y     1  _  e2 
If  e  <  1  this  is  an  ellipse  with  semi-axes 

if  e  >  1  it  is  a  hyperbola  with  semi-axes 

147.  Focus  and  Directrix.  The  distance  c  (in  absolute  value) 
from  the  center  0  to  the  focus  F^  is,  as  shown  above,  for  the 
ellipse  2 

c  =  — ^  =  ae, 


for  the  hyperbola 


c  =  ^  ^  ^  =  ae. 


The  distance  (in  absolute  value)  of   the  directrix  from  the 
center  0  is  for  the  ellipse,  since  g  =  a(l  —  e^)/e  =  a/e  —  ae  : 

e  e 

and  for  the  hyperbola,  since  q  =  ae  —  a/e : 

OD  =  c  —  q  =  ae  —  ae -{--  =  -' 
e      e 

It  is  clear  from  the  symmetry  of  the  ellipse  and  hyperbola 
that  each  of  these  curves  has  two  foci,  one  on  each  side  of  the 
center  at  the  distance  ae  from  the  center,  and  two  directrices 
whose  equations  are  a;  =  ±  a/e. 

EXERCISES 
1.   Sketch  the  following  conies : 

(«)'-  =  ;r-r-| 7'        (&)  r  =  ^  .^       ,         ic)r~        ^ 


2  +  3  cos  0  2  4-  cos  0  1  —  2  cos  0 


144  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  147 

2.  Sketch  the  following  conies  and  find  their  foci  and  directrices  : 

(a)  a;2  +  4  2/2  :=  4,  (b)  4  x2  +  y^  =  4, 

(c)  a:2  -  4  2/2  =  4,  (d)  4x^  -  y'  =  4, 

(e)  16  a;2  -\-2by^  =  400,  (/)  9  a;2  _  16  y2  =  144^ 

{g)  9  ic2  -  16  2/2  +  144  =  0,  (h)  x^-y^  =  2. 

3.  Show  that  the  following  equations  represent  ellipses  or  hyperbolas 
and  find  their  centers,  foci,  and  directrices : 

(a)  a;2  +  32/2_2x+ 62/4- 1  =  0,        (6)  12x2- 42/2  -  12x  -  9  =  0, 
(c)  5x2  +  2/2 +  20x  + 15  =  0,  (d)  5x2-42/2  +  82/4-16  =  0. 

4.  Find  the  length  of  the  latus  rectum  of  an  ellipse  and  a  hyperbola 
in  terms  of  the  serai-axes. 

5.  Show  that  when  tangents  to  an  ellipse  or  hyperbola  are  drawn 
from  any  point  of  a  directrix  the  line  joining  the  points  of  contact  passes 
through  a  focus. 

6.  From  the  definition  (§  141)  of  an  ellipse  and  hyperbola,  show  that 
the  sum  and  difference  respectively  of  the  focal  radii  of  any  point  of  the 
conic  is  constant. 

7.  Find  the  locus  of  the  midpoints  of  chords  drawn  from  one  end  of  : 
(a)  the  major  axis  of  an  ellipse  ;  (6)  the  minor  axis. 

8.  Find  the  locus  of  §  141  when  the  fixed  point  lies  on  the  fixed  line. 

148.  The  Conies  as  Sections  of  a  Cone.  As  indicated  by 
their  name  the  conic  sections,  i.e.  the  parabola,  ellipse,  and 
hyperbola,  can  be  defined  as  the  curves  in  which  a  right  circu- 
lar cone  is  cut  by  a  jjlane  (§  140). 

In  Figs.  83,  84,  85,  Fis  the  vertex  of  the  cone,  ^  CVC=2  a 
the  angle  at  its  vertex ;  OQ  indicates  the  cutting  plane,  CVC 
that  plane  through  the  axis  of  the  cone  which  is  perpen- 
dicular to  the  cutting  plane.  The  intersection  OQ  of  these 
two  planes  is  evidently  an  axis  of  symmetry  for  the  conic. 

The  conic  is  a  parabola,  ellipse,  or  hyperbola,  according 
as  OQ  is  parallel  to  the  generator  VC  of  the  cone  (Fig. 
83),  meets  VC  at  a  point  (7  belonging  to  the  same  half-cone 


VIII,  §  149] 


CONIC  SECTIONS 


145 


as  does  0  (Fig.  84),  or  meets  FC  at  a  point  (X  of  the  other 
half-cone  (Fig.  85).      If  the  angle 
GOQ  be  called  /8,  the  conic  is 

a  parabola  if  y8  =  2  a  (Fig.  83), 
an  ellipse  \i  p>2a  (Fig.  84), 
a  hyperbola  if  y8  <  2  «  (Fig.  85). 

In  each  of  the  three  figures  CC 
represents  the  diameter  2  r  of  any 
cross-section  of  the  cone  (i.e.  of  any 
section  at  right  angles  to  its  axis). 
We  take  0  as  origin,  OQ  as  axis  Fig.  83 

Oa;,  so  that  (Fig.  83)  OQ  =  x,  QP=y  are  the  coordinates  of 
any  point  P  of  the  conic. 

As  QP  is  the  ordinate  of  the  circular  cross-section  CPQ'P* 
we  have  in  each  of  the  three  cases  y^  =  QP^  =  CQ,  •  QC 

149.   Parabola.     In  the  first  case  (Fig.  83),  when  ^  =  2  a  so 
that  OQ  is  parallel  to  VQ\  the  expression 

X     OQ    oq   ^ 

is  constant,  i.e.  the  same  at  whatever  distance  from  the  vertex 
we  may  take  the  cross-section  CPQ'P' .  For,  QC"  is  equal  to 
the  diameter  OB  —  2r^  of  the  cross-section  through  0,  and 
CQ/OQ  =  CO/  VC=  2  r/r  esc  «  =  2  sin  a.  Hence,  denoting 
the  constant  r^  sin  a  by  j)  we  have 

||.QC'  =  4rosin«=4p. 

The  equation  of  the  conic  in  this  case,  referred  to  its  axis  OQ 
and  vertex  0,  is  therefore  y^  =  4:px.  Notice  that  asp  =  ^o  sin  a 
the  focus  is  found  as  the  foot  of  the  perpendicular  from 
the  midpoint  of  OB  on  OQ. 


146 


PLANE  ANALYTIC  GEOMETRY     [VIII,  §  150 


150.   Ellipse.      In    the    second    case   (Fig.    84),   i.e.   when 
yS  >  2  a,  if  we  put  '00'  =  2  a,  it  can  be  shown  that 

x{2a-x)      OQ'QO' 

is  constant.  For  we  have  QI^  = 
CQ  •  QC  and  from  the  triangles  CQOj 
QOO'j  observing  that 

-^  QaC  =l3-2a: 


whence 


C'(^  _        sm  /d                              / 

]>.      ^ 

OQ     sin(i,r-a)'                   /     '" 

-^  ->\ 

Q(7_sin(^-2a) 
Q(y     sin(i,r  +  a)' 

Fig.  84 

QP^      _sin^sin()8-2a) 

OQ .  Q(y  --2  ' 


cos'  a 


an  expression  independent  of  the  position  of  the  cross-section 
CC    Denoting  this  positive  constant  by  Zc*,  we  find  the  equation 

y^  =  fe(2a-a;),  or  (^~^)' 

which  is  an  ellipse,  with  semi-axes  a, 
ka,  and  center  (a,  0). 

151.  Hyperbola.  In  the  third  case 
(Fig.  85),  proceeding  as  in  the  second 
and  merely  observing  that  now  QC/ 
=  —  (2  a  -\-  x),  we  find  the  equation 

y^  =  k'^x  (2  a -\- x)  y 


I.e. 


a"  (kay       ' 


which   represents   a   hyperbola,   with 
semi-axes  a,  ka  and  center  (—a,  0). 


VIII,  §  152]  CONIC  SECTIONS  147 

152.  Limiting  Cases.  The  conic  is  an  ellipse,  hyperbola,  or 
parabola  according  as  ^8  >  2  a,  <  2  a,  or  =  2  a.  Hence  the 
parabola  can  be  regarded  as  the  limiting  case  of  either  an 
ellipse  or  a  hyperbola  whose  center  is  removed  to  infinity. 

If  when  /8  >  2  a  (Fig.  84),  we  let  /8  approach  tt,  or  if  when 
/8  <  2  a  (Fig.  85),  we  let  /?  approach  0,  the  cutting  plane  be- 
comes in  the  limit  a  tangent  plan^  to  the  cone.  It  then  has 
in  common  with  the  cone  the  points  of  the  generator  VC, 
and  these  only.  A  single  straight  line  can  thus  appear  as  a 
limiting  case  of  an  ellipse  or  hyperbola. 

Finally  we  obtain  another  class  of  limiting  cases,  or  cases  of 
degeneration,  of  the  conies  if,  in  any  one  of  the  three  cases, 
we  let  the  cutting  plane  pass  through  the  vertex  V  of  the 
cone.  In  the  first  case,  /8  =  2  a,  the  cutting  plane  is  then  tan- 
gent to  the  cone  so  that  the  parabola  also  may  degenerate  into 
a  single  straight  line.  In  the  second  case,  ^  >  2  a,  if  /3=^  ir, 
the  ellipse  degenerates  into  a  single  point,  the  vertex  V  of  the 
cone.  In  the  third  case,  /8  <  2  a,  if  /?  ^  0,  the  hyperbola  de- 
generates into  two  intersecting  lines.  The  terra  conic  section, 
or  conic,  is  often  used  as  including  these  limiting  cases. 

EXERCISES 

1.  For  what  value  of  /3  in  the  preceding  discussion  does  the  conic  be- 
come a  circle  ? 

2.  Show  that  the  spheres  inscribed  in  a  right  circular  cone  so  as  to 
touch  the  cutting  plane  (Figs.  83,  84,  86)  touch  this  plane  at  the  foci  of 
the  conic. 

3.  The  conic  sections  were  originally  defined  (by  the  older  Greek 
mathematicians,  in  the  time  of  Plato,  about  400  b.c.)  as  sections  of  a 
cone  by  a  plane  at  right  angles  to  a  generator  of  the  cone  ;  show  that  the 
section  is  a  parabola,  ellipse,  or  hyperbola  according  as  the  angle  2  a  at 
the  vertex  of  the  cone  is  =  |  tt,  <  ^  tt,  >  ^  tt. 


148  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  153 

PART   XL     REDUCTION   OF   GENERAL   EQUATION 

153.  Equations  of  Conies.  We  have  seen  in  the  two  pre- 
ceding chapters  that  by  selecting  the  coordinate  system  in  a  con- 
venient way  the  equation  of  a  parabola  can  be  obtained  in  the 
simple  form 

that  of  an  ellipse  in  the  form 

a^^b^-^' 
and  that  of  a  hyperbola  in  the  form 

a"     b-'       ' 

When  the  coordinate  system  is  taken  arbitrarily,  the  carte- 
sian equations  of  these  curves  will  in  general  not  have  this 
simple  form ;  but  they  will  always  be  of  the  second  degree. 
To  show  this  let  us  take  the  common  definition  of  these  curves 
(§  141)  as  the  locus  of  a  point  whose  distances  from  a  fixed 
point  and  a  fixed  line  are  in  a  constant  ratio.  With  respect  to 
any  rectangular  axes,  let  x^ ,  y^  be  the  coordinates  of  the  fixed 
point,  ax  +  6?/  +  c  =  0  the  equation  of  the  fixed  line,  and  e  the 
given  ratio.     Then  by  §§  9  and  42  the  equation  of  the  locus  is 

V(^-.0^  +  (,-,0^  =  e  .  «^±M±^, 

or,  rationalized: 

(X  -  x,y  +  {y-y,f  =  ^^  (ax-\-by-^cy. 
a  -j-  0 

It  is  readily  seen  that  this  equation  is  always  of  the  second 
degree;  i.e.  that  the  coefficients  of  x^,  y^,  and  xy  cannot  all 
three  vanish. 


VIII,  §155]    EQUATION  OF  SECOND  DEGREE  149 

154.  Equation  of  Second  Degree.  Conversely,  every  equa- 
tion of  the  second  degree,  i.e.  every  equation  of  the  form  (§47) 
(1)  Ax^ -{-2 Hxy -\- By""  +  2  Qx  +  2Fy -^  C  =  0, 
where  A,  H,  B  are  not  all  three  zero,  in  general  represents  a 
conic.  More  precisely,  the  equation  (1)  may  represent  an 
ellipse,  a  hyperbola,  or  a  parabola;  it  may  represent  two 
straight  lines,  different  or  coincident ;  it  may  be  satisfied  by 
the  coordinates  of  only  a  single  point;  and  it  may  not  be 
satisfied  by  any  real  point. 

Thus  each  of  the  equations 

ar^  _  3  2/2  =  0,  x?/  =  0 
evidently  represents  two  real  different  lines ;  the  equation 

aj2_2a;  +  l=0 
represents  a  single  line,  or,  as  it  is  customary  to  say,  two  coin- 
cident lines ;  the  equation 

a;2  +  2/'  =  0 
represents  a  single  point,  while 

is  satisfied  by  no  real  point  and  is  sometimes  said  to  represent 
an  "imaginary  ellipse." 

The  term  conic  is  often  used  in  a  broader  sense  (compare  §  152) 
so  as  to  include  all  these  cases ;  it  is  then  equivalent  to  the 
expression  "locus  of  an  equation  of  the  second  degree.'"' 

It  will  be  shown  in  the  present  chapter  how  to  determine 
the  locus  of  any  equation  of  the  form  (1)  with  real  coefficients. 
The  method  consists  in  selecting  the  axes  of  coordinates  so  as 
to  reduce  the  given  equation  to  its  most  simple  form. 

155.  Translation  of  Axes.  The  transformation  of  the 
equation  (1)  to  its  most  simple  form  is  very  easy  in  the  par- 
ticular case  when  (1)  contains  no  term  in  xy,  i.e.  when  11=0. 
Indeed  it  suffices  in  this  case  to  complete  the  squares  in  x  and  y 
and  transform  to  pO/TQ/lhl  axes. 


150  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  155 

Two  cases  may  be  distinguished: 

(a)  11=:  0,  A=^  Oj  B  ^0,  so  that  the  equation  has  the  form 

(2)  Ax''-\-By^-{-2Gx-\-2Fy-{-C=  0. 

Completing  the  squares  in  x  and  y  (§  48),  we  obtain  an  equation 

of  the  form 

A{x-hy  +  B(y-ky  =  K, 

where  ^  is  a  constant ;  upon  taking  parallel  axes  through  the 
point  (hy  k)  it  is  seen  that  the  locus  is  an  ellipse,  or  a  hyper- 
bola, or  two  straight  lines,  or  a  point,  or  no  real  locus,  accord- 
ing to  the  values  of  A^  B,  K. 

(h)  H=0,  and  either  jB=  0  or  ^4=0,  so  that  the  equation  is 

(3)  Aa!'  +  2Gx-\-2Fy-]-C=0,oT  By^  +  2Ox-{-2Fy+C=0. 

Completing  the  square  in  x  or  y,  we  obtain 

(x-hy=p(y-k),  or  (y -ky  =  q{x-h); 

with  (h,  k)  as  new  origin  we  have  a  parabola  referred  to  vertex 
and  axis,  or  two  parallel  lines,  real  and  different,  coincident,  or 
imaginary. 

It  follows  from  this  discussion  that  the  absence  of  the  term  in 
xy  indicates  that,  in  the  case  of  the  ellipse  or  hyperbola,  its  axes, 
in  the  case  of  the  parabola,  its  axis  and  tangent  at  the  vertex^  are 
parallel  to  the  aaxs  of  coordinates. 

EXERCISES 
1.  Beduce  the  following  equations  to  standard  forms  and  sketch  the 
loci  : 

(a)  2  1/2  -  3  a;  +  8  y  +  11  =  0,  (6)  ic^  +  4  y2  _  g  -,.  +  4  y  +  6  =  0, 

(c)  6  x2  +  3  2,2  _  4  a;  ^  2  y  +  1  =  0,       {d)  a;2  _  9  y2  _  6  x  +  18  y  =  0, 
(c)  9  x2  +  9  y2  _  36  x+6  y  +  10=0,       (/)  2  x2  -  4  y2  -f  4  x  +  4  y  -  1  =  0, 
{g)  x2  +  y2  -  2  X  +  2  y  +  3  =  0,  ih)  3  x2  -  6  x  +  y  +  6  =  0, 

(0  x2  -  y2  _  4aj  -  2  y  +  3  =  0,  (j)  2  x2  -  5  x  +  12  =  0, 

(A:)  2x2 -5  a; +  2  =  0,  (?)  y2-4y  +  4  =  0. 


VIII,  §156]    EQUATION  OF  SECOND  DEGREE 


151 


2.  Find  the  equation  of  each  of  the  following  conies,  determine  the 
axis  perpendicular  to  the  given  directrix,  the  vertices  on  this  axis  (by 
division-ratio),  the  lengths  of  the  semi-axes,  and  make  a  rough  sketch 
in  each  case : 

(a)  with  x  —  2  =  0  as  directrix,  focus  at  (6,  3),  eccentricity  |  ; 

(&)  with  3a;  +  4y  —  6  =  0as  directrix,  focus  at  (5,  4),  eccentricity  \  ; 

(c)  with  x  —  y  —  2  =  0  as  directrix,  focus  at  (4,  0),  eccentricity  |. 

3.  Find  the  axis,  vertex,  latus  rectum,  and  sketch  the  parabola  with 
focus  at  (2,  —  2)  and  2a:  —  3?/— 5  =  0  as  directrix  (see  Ex.  2). 

4.  Prove  the  statement  at  the  end  of  §  166. 

5.  Find  the  equation  of  the  ellipse  of  major  axis  5  with  foci  at  (0,  0) 
and  (3,  1). 

156.  Rotation  of  Axes.  If  the  right  angle  xOy  formed  by 
the  axes  Ox,  Oy  be  turned  about  the  origin  0  through  an 
angle  6  so  as  to   take  the  new  position  x^Oyi  (Fig.  86),  the 


w 


(40 


relation  between  the  old  coordinates  OQ  =  x,  QP  =  y  of  any 
point  P  and  the  new  coordinates  OQ^^x^,  QiP=yi  of  the 
same  point  P  are  seen  from  the  figure  to  be 

x  =  Xi  cos  0  —  yi  sin  9, 

y  =  x^  sin  0 -\- yi  cos  6. 

By  solving  for  x^,  y^,  or  again  from  Fig.  86,  we  find 

Xi  =  X  cos  $  -\-ysmOj 

yi  =  —  X  sin  6  -{-y  cos  6. 
If  the  cartesian  equation  of  any  curve  referred  to  the  axes 


152 


PLANE  ANALYTIC  GEOMETRY     [VIII,  §  156 


Ox,  Oy  is  given,  the  equation  of  the  same  curve  referred  to  the 
new  axes  Ox^ ,  Oy^  is  found  by  substituting  the  values  (4)  for 
X,  y  in  the  given  equation. 

157.   Translation  and  Rotation.     To  transform  from  any- 
rectangular   axes  Ox,  Oy  (Fig.  87)  to  any  other  rectangular 


y 

1 

k 

h 

^  i     . 

0 

ji 

Fia.  87 

axes   OiiCi,    Oi2/i,  we  have   to  combine  the  translation   00^ 
(§  13)  with  the  rotation  through  an  angle  B  (§  156). 

This  can  be  done  by  first  transforming  from  Ox,  Oy  to  the 
parallel  axes  Oix\  Oiy'  by  means  of  the  translation  (§  13) 

x  =  x'  -{-  h, 
y  =  y'-\-k, 
and  then  turning  the  right  angle   x'Oiy'  through  the  angle 
$  =  x'OiXi,  which  is  done  by  the  transformation  (§  156) 
x'  =  Xi  cos  6  —  yi  sin  0, 
y'  ==  Xi  sin  0-^yi  cos  6. 
Eliminating  x',  y',  we  find 

x  =  Xi  cos  6 


(5) 


2/i  sin  0-\-hy 
,y  =  Xi  sin  6 -\- yi  cos  6 -\- k. 
The  same  result  would  have  been  obtained  by  performing 
first  the  rotation  and  then  the  translation. 

It  has  been  assumed  that  the  right  angles  xOy  and  x^Oy^  are 
superposable ;  if  this  were  not  the  case,  it  would  be  necessary 
to  invert  ultimately  one  of  the  axes. 


VIII,  §  157]    EQUATION  OF  SECOND  DEGREE  153 

EXERCISES 

1.  Find  the  coordinates  of  each  of  the  following  points  after  the  axes 
have  been  rotated  about  the  origin  through  the  indicated  angle  : 

(a)    (3,  4),  iT.  (6)    (0,  5),-|7r. 

(c)    (-3,  2),  e?  =  tan-i|.  (d)    (4,  -3),  ^tt. 

2.  If  the  origin  is  moved  to  the  point   (2,  —  1)  and  the  axes  then 

rotated  through  30°,  what  will  be  the  new  coordinates  of  the  following 

points  ? 

(a)    (0,0).  (6)    (2,3).  (c)    (6,-1). 

3.  Find  the  new  equation  of  the  parabola  y^  =  4  ax  after  the  axes  have 
been  rotated  through :     (a)    ^tt     ,     (&)    Jtt    ,     (c)    ir    . 

4.  Show  that  the  equation  x"^  +  y'^  =  «^  is  not  changed  by  any  rotation 
of  the  axes  about  the  origin.     Why  is  this  true  ? 

5.  Find  the  center  of  the  circle  (x—  a)^  +  y'^  =a^  after  the  axes  have 
been  turned  about  the  origin  through  the  angle  d.  What  is  the  new 
equation  ? 

6.  For  each  of  the  following  loci  rotate  the  axes  about  the  origin 
through  the  indicated  angle  and  find  the  new  equation : 

(a)   x2-y2  +  2  =  o,  Itt.  (b)   x^-y^  =  aMTr. 

(c)   y  =  mx  +  6,  ^  =  tan-i m.      (d)    12a;2  -  7xy  -  12i/2  =  0,  ^  =  tan-i|. 

(^)    -2  +  !^=^'^'^-  ^^^    x2_2/2  =  0,i,r. 

a2      0^ 

7.  Through  what  angle  must  the  axes  be  turned  about  the  origin  so 
that  the  circle  a;2  +  i/2_3x  +  4?/  —  5  =  0  will  not  contain  a  linear  term 
in  x? 

8.-  Suppose  the  right  angle  XiOyi  (Fig.  89)  turns  about  the  origin  at 
a  uniform  rate  making  one  complete  revolution  in  two  seconds.  The 
coordinates  of  a  point  with  respect  to  the  moving  axes  being  (2,  1),  what 
are  its  coordinates  with  respect  to  the  fixed  axes  xOy  at  the  end  of  ; 
(a)  i  sec.  ?     (&)   I  sec.  ?    (c)   1  sec.  ?     (d)    1|  sec.  ? 

9.  In  Fig.  89,  draw  the  line  OP,  and  denote  Z  QOP  by  </).  Divide 
both  sides  of  each  of  the  equations  (4)  by  OP  and  show  that  they  are 
then  equivalent  to  the  trigonometric  formulas  for  cos  (^  +  0)  and 
sin  (0  +  0). 


154  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  158 

158.  Removal  of  the  Term  in  x^y.  The  general  equation 
of  the  second  degree  (1),  §  154,  when  the  axes  are  turned  about 
the  origin  through  an  angle  ^  (§  156),  becomes : 

A{^^  cos  0—y\  sin  BY 

■\-2H{x^  cos  B  —  y^  sin  B)  {x^  sin  B-\-yx  cos  B) 

+  B  (x'l  sin  ^  +  2/i  cos  BJ- 

+  2  G^(£Ci  cos ^  —  2/i  sin  B) 

+  2  F{x^  sin  ^  +  2^1  cos  ^)  +  O  =  0. 

This  is  an  equation  of  the  second  degree  in  x^  and  ^i  in 
which  the  coefficient  of  x^y^  is  readily  seen  to  be 

—  2ulcos^sin^  +  25sin^cos^H-2ir(cos2^-sin2^) 

=  (5-^)sin2^  +  2ircos2^. 

It  follows  that  if  the  axes  be  turned  about  the  origin 
through  an  angle  B  such  that 

(5-^)sin2^  +  2^cos2d  =  0, 
•i.e.  such  that 
(6)  tan2e  =  ^^, 

the  equation  referred  to  the  new  axes  will  contain  no  term  in 
a^?/i  and  can  therefore  be  treated  by  the  method  of  §  155. 
According  to  the  remark  at  the  end  of  §  155  this  means 
that  the  new  axes  OaJi,  Oyi,  obtained  by  turning  the  original 
axes  Oa;,  Oy  through  the  angle  B  found  from  (6),  are  parallel 
to  the  axes  of  the  conic  (or,  in  the  case  of  the  parabola,  to  the 
axis  and  the  tangent  at  the  vertex). 

The  equation  (6)  can  therefore  be  used  to  determine  ttie 
directions  of  the  axes  of  the  conic;  but  the  process  just  indicated 
is  generally  inconvenient  for  reducing  a  numerical  equation  of 
the  second  degree  to  its  most  simple  form  since  the  values  of 
cos  B  and  sin  B  required  by  (4)  to  obtain  the  new  equation  are 
in  general  irrational. 


VIII,  §  159]    EQUATION  OF  SECOND  DEGREE  155 

EXERCISES 

1.  Through  what  angle  must  the  axes  be  turned  about  the  origin  td 
remove  the  term  in  xy  from  each  of  the  following  equations  ? 

(a)  3a:2+2V'3x2/+2/'^-3a;+4?/-10=0.     (6)  x2 +  2V3a;y  +  7y2_i5=:0. 
(c)   2a;2-3x2/  +  2?/2  +  x-?/  +  7=0.         {d)xy  =  2a'^. 

2.  Reduce  each  of  the  following  equations  to  one  of  the  forms  in  §  244. 
{a)xy=-2.  (b)   Qx"^- 6xy  -  6y^  =  0. 

(c)   3a;2-10xy +  3?/2  +  8  =  0.  (d)    VSx^  -  lOxy  +  ISy'^  -  72  =  0. 

159.  Transformation  to  Parallel  Axes.  To  transform  the 
general  equation  of  the  second  degree  (1),  §  154,  to  parallel 
axes  through  any  point  (x^,  y^j  we  have  to  substitute  (§  13) 

x  =  x'^Xq,     2/=2/H-2/oj 

the  resulting  equation  is 

A7f^  +  2  Hxfy'  +  By'^  +  2  {Ax,  +  Hy,  +  G)  a/ 

+  2{Hx,  +  By,  +  F)y'^-C^0, 

where  the  new  constant  term  is 

(7)  C'  =  Ax,^^-2Hx^,^By,'-\-2  0x,^.2Fy,+a 

It  thus  appears  that  after  any  translation  of  the  coordinate 
system : 

(a)  the  coefficients  of  the  terms  of  the  second  degree  remain 
unchanged ; 

(6)  the  new  coefficients  of  the  terms  of  the  first  degree  are 
linear  functions  of  the  coordinates  of  the  new  origin ; 

(c)  the  new  constant  term  is  the  result  of  substituting  the 
coordinates  of  the  new  origin  in  the  left-hand  member  of  the 
original  equation. 


156  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  160 

160.  Transformation  to  the  Center.  The  transformed  equa- 
tion will  contain  no  terms  of  the  first  degree,  i.e.  it  will  be  of 
the  form 

(8)  Ax'^  +  2  Ilxy  -f  By'^  +  C"  =  0, 

if  we  can  select  the  new  origin  (a;,,,  y^)  so  that 

.^.  Ax,  +  Hy,+  G  =  0, 

^^  Hx,  +  By,  +  F  =  0. 

This  is  certainly  possible  whenever 

A     H 


and  we  then  find  : 

nm  _FH-GB       _  GH-  FA 

^^^  '~AB-H-''^'~  AB-H^' 

As  the  equation  (8)  remains  unchanged  when  x',  ]/  are 
replaced  by  —  ic',  —  y' ^  respectively,  the  new  origin  so  found  is 
the  center  of  the  curve  (§  133).  The  locus  is  therefore  in 
this  case  a  central  conic,  i.e.  an  ellipse  or  a  hyperbola;  but  it 
may  reduce  to  two  straight  lines  or  to  a  point  (see  §  162).  It 
might  be  entirely  imaginary,  viz.  if  H=  0 ;  but  the  case  when 
H=0  has  already  been  discussed  in  §  155. 

We  shall  discuss  in  §  164  the  case  in  which  AB  —  H^  =  0. 

161.  The  Constant  Term  and  the  Discriminant.  The  cal- 
culation of  the  constant  term  C  can  be  somewhat  simplified 
by  observing  that  its  expression  (7)  can  be  written 

C'=(Ax,  +  Hy,  +  G)x,  +  (Hx,-^By,  +  F)y,-\-Gx,-\-Fy,-}-C, 

i.e.,  owing  to  (9), 

(11)  C'=Gx,-{-Fy,-\-a 

If  we  here  substitute  for  x^,  y^  their  values  (10)  we  find  : 

C  =  G^^//  -  G^B  +  FGH  -  F^A  +  ABO-H^O 
AB-H^ 


VIII,  §  163]    EQUATION  OF  SECOND  DEGREE  157 

The  numerator,  which  is  called  the  discriminant  of  the  equa- 
tion of  the  second  degree  and  is  denoted  by  D,  can  be  written 
in  the  form  of  a  symmetric  determinant,  viz. 

A     H    O 

D==  H    B     F 

G     F     C 

If  we  denote  the  cofactors  of  this  determinant  by  the  corre- 
sponding small  letters,  we  have  XQ  =  g/c,  y^^f/c,  C/=D/c- 
Notice  that  the  coefficients  of  the  equations  (9),  which  deter- 
mine the  center,  are  given  by  the  first  two  rows  of  D,  while  the 
third  row  gives  the  coefficients  of  C  in  (11). 

162.  Straight  Lines.  After  transforming  to  the  center,  i.e. 
obtaining  the  equation  (8),  we  must  distinguish  two  cases 
according  as  C"  =  0  or  C  =f=  0.  The  condition  C"  =  0  means 
by  (7)  that  the  center  lies  on  the  locus ;  and  indeed  the  homo- 
geneous equation 

Ax"  +  2Hx'y'-{-By"  =  0 

represents  two  straight  lines  through  the  new  origin  (Xq  ,  2/0) 
(§  45).  The  separate  equations  of  these  lines,  referred  to 
the  new  axes,  are  found  by  factoring  the  left-hand  member. 
As  we  here  assume  (§  160)  that  AB—H'^=^0,  and  H=^0,  the 
lines  can  only  be  either  real  and  different,  or  imaginary.  In 
the  latter  case  the  point  (o^,, ,  Vo)  is  the  only  real  point  whose 
coordinates  satisfy  the  original  equation. 

163.  Ellipse  and  Hyperbola.  If  C'=^0  we  can  divide  (8) 
by  —  C  so  that  the  equation  reduces  to  the  form 

(12)  ax'+2hxy-^by^  =  l. 

This  equation  represents  an  ellipse  or  a  hyperbola  (since  we 
assume  h^O).  The  axes  of  the  ellipse  or  hyperbola  can  be 
found  in  magnitude  and  direction  as  follows. 


158  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  163 

If  an  ellipse  or  hyperbola,  with  its  center,  be  given  graphi- 
cally, the  axes  can  be  constructed  by  inter- 
secting the  curve  with  a  concentric  circle 
and  drawing  the  lines  from  the  center  to 
the  intersections;  the  bisectors  of  the 
angles  between  these  lines  are  evidently 
the  axes  of  the  curve  (Fig.  88). 

The  intersections  of  the  curve  (12)  with 
a  concentric  circle  of  radius  r  are  given  by 
the  simultaneous  equations 

ax'-ir2hxy  +  hf  =  l,  x'+if  =  i''', 

dividing  the  second  equation  by  r^  and  subtracting  it  from  the 
first,  we  have 

(13)  (« -^)»^  + 2 'ixZ' +(6-^)2/^  =  0. 

This  homogeMeous  equation  represents  two  straight  lines 
through  the  origin,  and  as  the  equation  is  satisfied  by  the 
coordinates  of  the  points  that  satisfy  both  the  preceding  equar 
tions,  these  lines  must  be  the  lines  from  the  origin  to  the  inter- 
sections of  the  circle  with  the  curve  (12).  If  we  now  select  r 
so  as  to  make  the  two  lines  (13)  coincide,  they  will  evidently 
coincide  with  one  or  the  other  of  the  axes  of  the  curve  (12). 
The  condition  for  equal  roots  of  the  quadratic  (13)  in  y/x  is 

(14)  (a-r^(b-r^-n^^o. 

This  equation,  which  is  quadratic  in  1/r*  and  can  be  written 

(14')  (JJ-(«  +  6)i  +  «6-'i'  =  0. 

determines  the  lengths  of  the  axes.  If  the  two  values  found  for 
r^  are  both  positive,  the  curve  is  an  ellipse ;  if  one  is  positive 


VIII,  §  164]    EQUATION  OF  SECOND  DEGREE  159 

and  the  other  negative,  it  is  a  hyperbola ;  if  both  are  negative, 
there  is  no  real  locus. 

Each  of  the  two  values  of  l/r^  found  from  (14'),  if  substi- 
tuted in  (13),  makes  the  left-hand  member,  owing  to  (14),  a 
complete  square.     Tlie  equations  of  the  axes  are  therefore 


or,  multiplying  by  V a  —  l/r^  and  observing  (14) 

\x±hy  =  0. 


(--i) 


164.  Parabola.  It  remains  to  discuss  the  case  (§  160)  of  the 
general  equation  of  the  second  degree. 

Ax"  +  2 Hxy  -{-  By""  +  2  Ox  +  2Fy  -^  C  =  0, 
in  which  we  have  ^^  —  H^  =  Q 

This  condition  means  that  the  terms  of  the  second  degree  form 
a  perfect  square  : 

Ax^  +  2  Hxy  -\-  By^  =  (VAx  +  VBy^, 
Putting  ^A  =  a  and  V-B  =  6  we  can  write  the  equation  of  the 
second  degree  in  this  case  in  the  form 

(15)  {ax  +  hyf  =  -2Gx-2Fy-G, 

If  O  and  F  are  both  zero,  this  equation  represents  two  parallel 
straight  lines,  real  and  different,  real  and  coincident,  or  im- 
aginary according  as  0  <  0,  C  =  0,  O  >  0. 

If  G  and  F  are  not  both  zero,  the  equation  (15)  can  be  inter- 
preted as  meaning  that  the  square  of  the  distance  of  the  point 
{x,  y)  from  the  line 

(16)  ax  +  by  =  0 

is  proportional  to  the  distance  of  (x,  y)  from  the  line 

(17)  2Gx  +  2Fy-^C=0. 

Hence  if  these  lines  (16),  (17)  happen  to  be  at  right  angles,  the 


160  PLANE  ANALYTIC  GEOMETRY     [VIII,  §  164 

locus  of  (15)  is  a  parabola,  having  the  line  (16)  as  axis  and  the 
line  (17)  as  tangent  at  the  vertex. 

But  even  when  the  lines  (16)  and  (17)  are  not  at  right  angles 
the  equation  (15)  can  be  shown  to  represent  a  parabola.  For 
if  we  add  a  constant  k  within  the  parenthesis  and  compensate 
the  right-hand  member  by  adding  the  terms  2  akx  +  2  bky  +  k^, 
the  locus  of  (15)  is  not  changed ;  and  in  the  resulting  equation 

(18)  (ax  +  by  +  ky  =  2{ak  -  G)x  +  2{bk  -F)y  +  k^-G 
we  can  determine  k  so  as  to  make  the  two  lines 

(19)  ax-\-by-^k  =  0, 

(20)  2(ak -  G)x  +  2{bk  -F)y  +  k^-C=0 

perpendicular.    The  condition  for  perpendicularity  is 

a{ak-G)  +  b{bk-F)=0, 
whence 

(21)  k  =  ^^±^. 

With  this  value  of  k,  then,  the  lines  (19),  (20)  are  at  right 
angles ;  and  if  (19)  is  taken  as  new  axis  Ox  and  (20)  as  new 
axis  Oy,  the  equation  (18)  reduces  to  the  simple  form 

y^  =  px. 
The  constant  p,  i.e.  the  latus  rectum  of  the  parabola,  is  found 
by  writing  (18)  in  the  form 
/ax  -{-by-\-  fey 


2 V(afc  -  Gy  +  (bk  -  Fy  2(ak  -  G)x  +  2(bk-F)y-^k^- C, 

a^+b^  2V(ak-Gy-\-{bk-Fy 

hence 

^  =  ^^^  V(aA:  -GO*  +  (bk  -  Fy. 

Substituting  for  k  its  value  (21)  we  can  reduce  it  to 
^_2(aF-bCf) 
(a2  +  62)i 


VIII,  §  164]    EQUATION  OF  SECOND   DEGREE  161 

EXERCISES 

1.  Find  the  equation  of  each  of  the  following  loci  after  transforming 
to  parallel  axes  through  the  center : 

(a)  Sx'^-ixy-y^-Sx-iy+7  =  0. 
(6)  5  x2  +  6  x?/  +  ?/^  +  6  X  -  4  y  —  5  =  0. 

(c)  2  x2  +  xy  -  6  y2  _  7  a;  —  7  ?/  +  5  =  0. 

(d)  x2  -  2  xy  -  2/2  +  4  X  -  2  ?/  -  8  =  0. 

2.  Find  that  diameter  of  the  conic  3x^  —  2xy—4:y^-{-6x—^y  ■{■2=0 
(a)  which  passes  through  the  origin,  (&)  which  is  parallel  to  each  co- 
ordinate axis. 

3.  For  what  values  of  k  do  the  following  equations  represent  straight 
lines  ?    Find  their  intersections. 

(a)  2x^-xy-Sy^-6x  +  l9y  -\-k  =  0. 
(&)  kx^  +  2  xy  +  y"^  -  X  -  y  -  6  =  0. 

(c)  3  x2  -  4  xi/  +  ^•^/2  +  8  2/  -  3  =  0. 

(d)  X*  +  2  ?/2  +  6  X  -  4  y  +  A;  =  0. 

4.  Show  that  the  equations  of  conjugate  hyperbolas  x^/a^—y^/b^=±l 
and  their  asymptotes  x^/a'^—y^/b^  =  0,  even  after  a  translation  and  rota- 
tion of  the  axes,  will  differ  only  in  the  constant  terms  and  that  the  con- 
stant term  of  the  asymptotes  is  the  arithmetic  mean  between  the  constant 
terms  of  the  conjugate  hyperbolas. 

6.  Find  the  asymptotes  and  the  hyperbola  conjugate  to 
2  x2  -  xy  -  15  2/2  +  X  +  19  y  +  16  =  0. 

6.  Find  the  hyperbola  through  the  point  (—2,  1)  which  has  the  lines 
2x  —  y+l=0,  3x  +  2?/  —  6  =  0  as  asymptotes.  Find  the  conjugate 
hyperbola. 

7.  Show  that  the  hyperbola  xy  =  a^  is  referred  to  its  asymptotes  as 
coordinate  axes.  Find  the  semi-axes  and  sketch  the  curve.  Find  and 
sketch  the  conjugate  hyperbola. 

8.  The  volume  of  a  gas  under  constant  temperature  varies  inversely 
as  the  pressure  (Boyle's  law),  i.e.  vp  =  c.  Sketch  the  curve  whose  ordi- 
nates  represent  the  pressure  as  a  function  of  the  volume  for  different 
values  of  c  ;  e.g.  take  c  =  1,  2,  3. 

9.  Sketch  the  hyperbola  (x  —  a)(y  —  b)  =  c^  and  its  asymptotes.  In- 
terpret the  constants  a,  6,  c  geometrically. 


162  PLANE  ANALYTIC  GEOMETRY    [VIII,  §  164 

10.  Sketch  the  hyperbola  xy-\-3y  —  6  =  0  and  its  asymptotes. 

11.  Find  the  center  and  semi-axes  of  the  following  conies,  write  their 
equations  in  the  most  simple  form,  and  sketch  the  curves : 

(a)  6  x^  -  6  xy  +  6  y^  +  12V2  z  -  W2y  +  8  =  0. 

(6)  a;2  -  6  V3  xy  -  6  y2  _  16  =  0.     (c)  x^  +  xy  +  y^  _  3  y  +  6  =  0. 

(d)  13  x2  -  6V3xy  +  7  y^  _  64  =  0. 

(e)  2  x2  -  4  xy  +  y2  _|_  2  x  -  4  2/  -  f  =  0. 
(/)  3x2  +  2xy  +  y2  +  6x  +  4y  +  ^  =  0. 

IS.   Sketch  the  following  parabolas  : 

(a)  x2  -  2VSxy  -\-Sy^-  eVSx  -  6  y  =  0. 

(6)  x2  -  6  xy  +  9  y2  _  3  X  +  4y  -  1  =  0. 

IS.  Show  that  the  following  combinations  of  the  coeflBcients  of  the 
general  equation  of  the  second  degree  are  invariants  {i.e.  remain  un- 
changed) under  any  transformation  from  rectangular  to  rectangular  axes : 

(a)  A-hB.  (6)  AB  -  H\  (c)  {A  -5)2  +  4  m. 

14.  Show  that  x2  +  y 2  =  a^  represents  a  parabola.     Sketch  the  locus. 

15.  Find  the  parabola  with  x  +  y  =  0  as  directrix  and  (^  a,  J  a)  as 
focus. 

16.  Let  five  points  A^  J5,  C,  D^  E  be  taken  at  equal  intervals  on  a 
line.  Show  that  the  locus  of  a  point  P  such  that  AP  •  EP  =  BP  •  DP  is 
an  equilateral  hyperbola.     (Take  C  as  origin.) 

17.  The  variable  triaYigle  AQB  is  isosceles  with  a  fixed  base  AB. 
Show  that  the  locus  of  the  intersection  of  the  line  AQ  with  the  perpen- 
dicular to  QB  through  B  is  an  equilateral  hyperbola. 

18.  Let  ^  be  a  fixed  point  and  let  Q  describe  a  fixed  line.  Find  the 
locus  of  the  intersection  of  a  line  through  Q  perpendicular  to  the  fixed 
line  and  a  line  through  A  perpendicular  to  A  Q. 

19.  Find  the  locus  of  the  intersection  of  lines  drawn  from  the  extrem- 
ities of  a  fixed  diameter  of  a  circle  to  the  ends  of  the  perpendicular 
chords. 

20.  Show  by  (14'),  §163,  that  if  the  equation  of  the  second  degree 
represents  an  ellipse,  parabola,  hyperbola,  we  have,  respectively, 

AB-H^  >0,  =  0,  < p. 


CHAPTER  IX 

HIGHER  PLANE   CURVES 

PART  I.    ALGEBRAIC   CURVES 

165.  Cubics.  It  has  been  shown  (§  30)  that  every  equation 
of  the  first  degree, 

Oo 
+  a^x  +  6^2/  =  0, 

represents  a  straight  line;  and  (§  154)  that  every  equation  of 
the  second  degree, 

ao 
+  a^x  +  h^y 
+  a^^  +  ^^y  +  C22/2  =  0, 
either  represents  a  conic  or  is  not  satisfied  by  any  real  points. 
The  locus  represented  by  an  equation  of  the  third  degree, 

ao 

+  a^x  -f  h^ 

4-  ttgoj^  +  h^y  +  Cay* 

-I-  a^  +  h^^y  +  c^xy'^  +  d^^=  0, 

I.e.  the  aggregate  of  all  real  points  whose  coordinates  x,  y  satisfy 
this  equation,  is  called  a  cubic  curve. 

Similarly,  the  locus  of  all  points  that  satisfy  any  equation  of 
the  fourth  degree  is  called  a  quartic  ctirve;  and  the  terms  quintiCj 
sextic,  etc.,  are  applied  to  curves  whose  equations  are  of  the 
Jifthf  sixth,  etc.,  degrees. 

Even  the  cubics  present  a  large  variety  of  shapes;  still 
more  so  is  this  true  of  higher  curves.  We  shall  not  discuss 
such  curves  in  detail,  but  we  shall  study  some  of  their  properties. 

163 


164  PLANE  ANALYTIC  GEOMETRY       [IX,  §  166 

166.   Algebraic  Curves.     The  general  form  of  an  algebraic 
equation  of  the  nth  degree  in  x  and  y  is 

«o 
4-  a^x  +  h^ 
(1)  +a^-\-  b^y  -f  C22/2 

■^a^-\-b^y-\-c^'^-\-d^ 


+  a^a;"  +  b^x^'-^y  -\ \-Kxy''~^+  Z^"  =  0. 

The  coefficients  are  supposed  to  be  any  real  numbers,  those  in 
the  last  line  being  not  all  zero.  The  number  of  terms  is  not 
more  than  1  -f  2  -f  3  +  ...  -^(n  +  1)  =  i(n  +  l)(n  +  2). 

If  the  cartesian  equation  of  a  curve  can  be  reduced  to  this 
form  by  rationalizing  and  clearing  of  fractions,  the  curve  is 
called  an  algebraic  curve  of  degree  n. 

An  algebraic   curve   of  degree  n  can  be  intersected  by  a 

straight  line, 

Ax  +  By-{-C=0, 

in  not  more  than  n  points.  For,  the  substitution  in  (1)  of  the 
value  of  y  (or  of  x)  derived  from  the  linear  equation  gives  an 
equation  in  x  (or  in  y)  of  a  degree  not  greater  than  n ;  this 
equation  can  therefore  have  not  more  than  n  roots,  and  these 
roots  are  the  abscissas  (or  ordinates)  of  the  points  of  intersec- 
tion. 

We  have  already  studied  the  curves  that  represent  the  poly- 
nomial function 

y=aQ-{-a^x-\-a^-\ \-  a^iC* ; 

such  a  curve  is  an  algebraic  curve,  but  it  is  readily  seen  by 
comparison  with  the  preceding  equation  that  this  equation  is 
of  a  very  special  type,  since  it  contains  no  term  of  higher  de- 
gree than  one  in  y.  Such  a  curve  is  often  called  a  parabolic 
curve  of  the  nth  degree. 


IX,  §  169] 


ALGEBRAIC  CURVES 


165 


167.  Transformation  to  Polar  Coordinates.  The  cartesian 
equation  (1)  is  readily  transformed  to  polar  coordinates  by  sub- 
stituting 

x  =  r  cos  <^,     2/  =  ^  sin  <^ ; 

it  then  assumes  the  form : 

+  (ai  cos  <^  +  &i  sin  ^)r 
(2)  +  (as  cos^  <^  +  62  cos  </)  sin  <^  +  Cg  sin''  ^r^ 

+  (ag  cos^  <^  H-  63  cos^  <^  sin  <^  -h  Cg  cos  </>  sin^  <}!>  +  c^g  sin'  c^)/^ 


4-  (a„  cos"  <^  +  6„  cos''"^  <^  sin  <^  -f- 


+A;„cos  </>sin"-^  <^+Z„sin'»<^)r'" 
=  0. 


If  any  particular  value  be  assigned  to  the  polar  angle  <^,  this 
becomes  an  equation  in  r  of  a 
degree  not  greater  than  n.     Its 


roots  ri, 


•  represent  the  in- 


tercepts OPi,  OP2,  •••  (Fig.  89) 
made  by  the  curve  (2)  on  the  line 
y  =  tan  <f>  -  x.  Some  of  these 
roots  may  of  course  be  imaginary, 
and  there  may  be  equal  roots. 


Fig.  89 


168.  Curve  through  the  Origin.  The  equation  in  r  has  at 
least  one  of  its  roots  equal  to  zero  if,  and  only  if,  the  constant 
term  Uq  is  zero.  Thus,  the  necessary  and  sufficient  condition  that 
the  origin  0  be  a  point  of  the  curve  is  ao  =  0. 

This  is  of  course  also  apparent  from  the  equation  (1)  which 
is  satisfied  by  aj  =  0,  2/  =  0  if ,  and  only  if,  Oq  =  0. 

169.  Tangent  Line  at  Origin.  The  equation  (2)  has  at 
least  two  of  its  roots  equal  to  zero  if  ao=0  and  a^  cos  <^  + 
61  sin  0  =  0.     If  tti  and  61  are  not  both  zero,  the  latter  condition 


166  PLANE  ANALYTIC  GEOMETRY       [IX,  §  169 

can  be  satiefied  by  selecting  the  angle  <^  properly,  viz.  so  that 

tan<^  =  -^. 

The  line  through  the  origin  inclined  at  this  angle  <^  to  the 
polar  axis  is  the  tangent  to  the  curve  at  the  origin  0  (Fig.  90) . 
Its  cartesian  equation  is  y  =  tan  <^  •  a;  =  —  (ai/bi)x,  i.e. 
(3)  a,x  +  b,y  =  0. 

Thus,  if  ao  =  0  while  Oj ,  bi  are  not  both  zero,  the  curve  has 
at  the  origin  a  single  tangent ;  the  origin  0  is  therefore  called 
a  simple,  or  ordinary,  point  of  the  curve. 
In  other  words,  if  the  lowest  terms  in 
the  equation  (1)  of-  an  algebraic  curve 
are  of  the  first  degree,  the  origin  is  a 
simple  point  of  the  curve,  and  the  equor 
tion  of  the  tangent  at  the  origin  is  ob- 
tained by  equating  to  zero  the  terms  of 
the  first  degree. 

170.  Double  Point.  The  condition  ajcos  <^+6i  sin  <^  =  0 
necessary  for  two  zero  roots  is  also  satisfied  if  aj  =  0  and  6i  =  0; 
indeed,  it  is  then  satisfied  whatever  the  value  of  </>.  Hence,  if 
ao  =  0,  Oj  =  0,  6i  =  0,  the  equation  (2)  has  at  least  two  zero 
roots  for  any  value  of  <\>.  If  in  this  case  the  terms  of  the 
second  degree  in  (1)  do  not  all  vanish,  the  curve  is  said  to 
have  a  double  point  at  the  origin.  Thus,  the  origin  is  a  double 
point  if,  and  only  ify  the  lowest  terms  in  the  equation  (1)  are  of 
the  second  degree. 

171.  Tangents  at  a  Double  Point.  The  equation  (2)  will 
have  at  least  three  of  its  roots  equal  to  zero  if  we  have  a©  =  0, 
Oi  =  0,  6i  =  0  and 

Oj  cos*  <^  +  62  cos  </>  sin  <^  -h  Cj  sin'  <^  =  0. 


IX,  §  172] 


ALGEBRAIC  CURVES 


167 


If  tta,  62J  C2  are  not  all  zero,  we  can  find  two  angles  satisfying 
this  equation  which  may  be  real  and  different,  or  real  and 
equal,  or  imaginary.  The  lines  drawn  at  these  angles  (if  real) 
through  the  origin  are  the  tangents  at  the  double  point. 

Multiplying  the  last  equation  by  r^  and  reintroducing  carte- 
sian coordinates  we  obtain  for  these  tangents  the  equation 


w 


a^"^  +  h^y  +  C22/^  =  0. 


Fig.  91 


Thus,  if  the  loivest  terms  in  the  equation  (1)  are  of  the  second 
degree^  the  origin  is  a  double  point,  and  these  terms  of  the  second 
degree  equated  to  zero  represent  the  tangents  at  the  origin. 

172.  Types  of  Double  Point,  (a)  If  the  two  lines  (4)  are 
real  and  different,  the  double  point  is 
called  a  node  or  crunode;  the  curve  then 
has  two  branches  passing  through  the 
origin,  each  with  a  different  tangent 
(Fig.  91).  J- 

(6)  If  the  lines  (4)  are  coincident,  i.e. 
if  a-fi?  +  bcfcy  -f-  c^y^  is  a  complete  square, 
the  double  point  is  called  a  cusp,  or  spinode;  the  curve  then 
has   ordinarily  two  real  branches  tangent  to 
one  and  the  same  line  at  the  origin  (Fig.  92 
represents  the  most  simple  case). 

(c)  If  the  lines  (4)  are  imaginary,  the 
double  point  is  called  an  isolated  point,  or 
an  acnode;  in  this  case,  while  the  coordi- 
nates 0,  0  of  the  origin  satisfy  the  equation 
of  the  curve,  there  exists  about  the  origin 
a  region  containing  no  other  point  of  the 
curve,  so  that  no  tangents  can  be  drawn 
through  the  origin  (Fig.  93). 


Fia.  92 


Fig.  93 


168  PLANE  ANALYTIC  GEOMETRY        [IX,  §  172 

It  should  be  observed  that,  for  curves  of  a  degree  above 
the  third,  the  origin  in  case  (b)  may  be  an  isolated  point ;  this 
will  be  revealed  by  investigating  the  higher  terms  (viz.  those 
above  the  second  degree). 

173.  Multiple  Points.  It  is  readily  seen  how  the  reasoning 
of  the  last  articles  can  be  continued  although  the  investigation 
of  higher  multiple  points  would  require  further  discussion. 
The  result  is  this :  If  in  the  equation  of  an  algebraic  curve,  when 
rationalized  and  cleared  of  fractions,  the  lowest  terms  are  of 
degree  k,  the  origiri  is  a  k-tuple  point  of  the  curve,  and  the  tan- 
gents at  this  point  are  given  by  the  terms  of  degree  k,  equated 
to  zero. 

To  investigate  whether  any  given  point  (Xi ,  2/j)  of  an  alge- 
braic curve  is  simple  or  multiple  it  is  only  necessary  to  trans- 
fer the  origin  to  the  point,  by  replacing  xhy  x-\-Xi  and  y  by 
y  -{■  yi,  and  then  to  apply  this  rule. 

EXERCISES 
1.  Determine  the  nature  of  the  origin  and  sketch  the  curves : 

(a)  y  =  x'^-2x.        (6)  x'^  =  iy-y^.  (c)  (x  +  a){y,  + a)  =  a'^. 

(d)  y2  =  a;2(4  -  x).     (c)  y^  =  3i^.  (/)   x^  +  ^2  =  ^^j. 

ig)y^  =  x^  +  3fi.         (A)  x8  -  3  aa;y  +  1/8  =  0.     (i)   x*-y*-^Qxy^  =  0. 

8.  Determine  the  nature  of  the  origin  and  sketch  the  curve  (y—x^y=x**, 
for:  (a)  n  =  l.        (6)  n  =  2.        (c)  n  =  3.        (d)  w  =  4. 

3.  Locate  the  multiple  points,  determine  their  nature,  and  sketch  the 
curves : 

(a)  y^  =  x{x  +  3)2.       (6)  (y  -  3)2  =  xK       (c)  (y  +  iy  =  (x-  3)8. 
(d)y8=(x  +  l)(x-l)2. 

4.  Sketch  the  curve  y^  =:{x  —  a')(x  —  b)(x  —  c)  and  discuss  the  multi- 
ple points  when : 

Ca)  0<a<b<c.   (6)  0<a<6  =  c.   (c)  0<a  =  b<c.   (d)  0<a  =  b  =  c. 


IX,  §175]  SPECIAL  CURVES  169 

PART  11.     SPECIAL   CURVES 

174.  Conchoid.  A  fixed  point  0  and  a  fixed  line  I,  at  the 
distance  a  from  0,  being  given,  the  radius  vector  OQ,  drawn  from 
0  to  every  point  Q  of  I,  is  produced  by  a  segment  QP=  b  of  con- 
stant length;  the  locus  of  Pis  called  the  conchoid  of  Nicomedes. 

For  0  as  pole  and  the  perpendicular  to  I  as  polar  axis  the 

equation  of  Hs  , 

ri  =  a/  cos  <^ ; 

hence  that  of  the  conchoid  is 

r ^  +  b. 

cos  <^ 

If  the  segment  QP  be  laid  off  in  the  opposite  sense,  we  obtain 

the  curve 

r  = by 

cos  <^ 

which  is  also  called  a  conchoid.  Indeed,  these  two  curves 
are  often  regarded  as  merely  two  branches  of  the  same 
curve.  Transforming  to  cartesian  coordinates  and  rationaliz- 
ing, we  find  the  equation 

which  represents  both  branches.  Sketch  the  curve,  say  for 
6  =  2  a,  and  for  b  =  a/2,  and  determine  the  nature  of  the  origin. 

175.  Limacon.  If  the  line  I  be  replaced  by  a  circle  and  the 
fixed  point  0  be  taken  on  the  circle,  the  locus  of  P  is  called 
Pascal's  limacon. 

For  0  as  pole  and  the  diameter  of  the  circle  as  polar  axis 
the  equation  of  the  circle,  of  radius  a,  is  ri  =  2  a  cos  <^ ;  hence 
that  of  the  limaQon  is  : 

r  =  2  a  cos  <f> -\- b. 


170 


PLANE  ANALYTIC  GEOMETRY        [IX,  §  175 


If  6  =  2  a,  the  curve  is  called  the  cardioid;  the  equation 

then  becomes 

r  =  4  a  cos^  ^  <f>. 

Sketch  the  limaQons  for  6  =  3a,  2a,  a;  transform  to  car- 
tesian coordinates  and  determine  the  character  of  the  origin. 

176.  Cissoid.  OC/  =  a  being  a  diameter  of  a  circle,  let  any 
radius  vector  drawn  from  0  meet  the  circle  and  its  tangent  at  O' 
at  the  points  Q,  D,  respectively;  if  on  this  radius  vector  we  lay- 
off OR  =  QD,  the  locus  of  R  is  called  the  cissoid  of  Diodes. 

With  0  as  pole  and  OCy  as  polar  axis,  we  have 

OD  =  a/cos  <f>,  OQ  =  a  cos  <^ ; 
the  equation  is  therefore 


/I  A        sin2 

r  =  af cos  <f>  )=  a 

\cos  </>  J        cos 

or  in  cartesian  coordinates 


<f> 


a? 


a—x 


Fio.  »4 


If  instead  of  taking  the  difference  of  the  radii  vectores  of  the 
circle  and  its  tangent  we  take  their  sum,  we  obtain  the  so-called 
companion  of  the  cissoid, 

r  =  a(cos  <^  +  sec  <^), 

.2a-ar 


I.e. 


r 


x  —  a 


Sketch  this  curve. 


177.  Versiera.  With  the  data  of  §  176,  let  us  draw  through 
Q  a  parallel  to  the  tangent,  through  D  a  parallel  to  the  diameter ; 
the  locus  of  the  point  of  intersection  P  of  these  parallels  is 
called  the  versiera  (wrongly  called  the  "  witch  of  Agnesi "). 


IX,  §  178] 


SPECIAL  CURVES 


171 


We  have  evidently  with  0  as  origin  and  00^  as  axis  Ox: 
x  =  a  cos^  <f>,        y  =  a  tan  <f>, 
whence  eliminating  <^ : 

x= 

2/2  +  a2 

If  we  replace  the  tangent  at  0'  by  any 
perpendicular   to    00'  (Fig.  95),  at  the 
distance  b  from  0,  we  obtain  the  curve 
x  =  a  cos^  (f>,        y  =  h  tan  ^y 
_     a¥ 

which  reduces  to  the  versiera  for  h  =  a. 

Sketch  the  versiera,  and  the  last  curve  for  6  =  i  a. 

178.   Cassinian  Ovals.     Lemniscate.    Two  fixed  points  Fj, 
F2  being  given  it  is  known  that  the  locus  of  a  point  P  is : 


Fig.  96 


(a)  a  circle  if  i^iP/i^^^P  =  const.  (Ex.  7,  p.  54); 
(6)  an  ellipse  if  F^P  +  F^P^  const.  (§  114) ; 
(c)  a  hyperbola  if  PiP- ^2^=  const.  (§  119). 


The  locus  is  called  a  Cassinian  oval  if  FiP  >  F2P  =  const.    If 


172 


PLANE  ANALYTIC   GEOMETRY        [IX,  §  178 


we  put  F1F2  =  2  a,  the  equation,  referred  to  the  midpoint  0 
between  Fi  and  F2  as  origin  and  OF2  as  axis  Oxj  is 

[(x  +  ay  +  /]  l{x  -  ay  +  y2]  =  k^. 

In  the  particular  case  when  k  —  a^  the  curve  passes  through 
the  origin  and  is  called  a  lemniscate.  The  equation  then  re- 
duces to  the  form 

^x^  +  y^y  =  2a%x'-y% 

which  becomes  in  polar  coordinates  r^  =  2  a^  cos  2  </>. 
Trace  the  lemniscate  from  the  last  equation. 

(Ito/  Cycloid.     The  common  cycloid  is  the  path  described  by 
any  point  Pofa  circle  rolling  over  a  straight  line  (Fig.  97). 


If  A  be  the  point  of  contact  of  the  rolling  circle  in  any  posi- 
tion, 0  the  point  of  the  given  line  that  coincided  with  the  point 
P  of  the  circle  when  P  was  point  of  contact,  it  is  clear  that 
the  length  OA  must  equal  the  arc  AP=a$,  where  a  is  the 
radius  of  the  circle,  and  6=  "^ACP,  the  angle  through  which 
the  circle  has  turned  since  P  was  at  0.  The  figure  then  shows 
that,  with  0  as  origin  and  OA  as  axis  Ox : 

X  =  OQ  =  aO  — a  sin  6,    y  =  a  —  a  cos  6. 

These  are  the  parameter  equations  of  the  cycloid.  The  curve  has 
an  infinite  number  of  equal  arches,  each  with  an  axis  of  sym- 
metry (in  Fig.  97,  the  line  x  =  wa)  and  with  a  cusp  at  each 
end.    Write  down  the  cartesian  equation. 


IX,  §  181] 


SPECIAL  CURVES 


173 


180.   Trochoid.     The  path  described  by  any  point  P  rigidly 
connected  with  the  rolling  circle  is  called  a  trochoid.     If  the 


Fig.  98. —The  Trochoids 
distance  of  P  from  the  center  C  of  the  circle  is  6,  the  equations 
of  the  trochoid  are  x=  aO—  b  sin  6,  y  =  a  —  b  cos  0. 
Draw  the  trochoid  for  b  =  ^a  and  for  6  =  |  a. 

181.  Epicycloid.  The  path  described  by  any  point  P  of  a 
circle  rolling  on  the  outside  of  a  fixed  circle  is  called  an  epicy- 
cloid (Fig.  99). 

Let  0  be  the  center,  b  the 
radius,  of  the  fixed  circle,  Cthe 
center,  a  the  radius,  of  the  rolling 
circle;  and  let  Aq  be  that  point 
of  the  fixed  circle  at  which  the 
describing  point  P  is  the  point 
of  contact.  Put  A^OA  =  <f>,  ACP 
=  6.  As  the  arcs  AAq  and  AP 
are  equal,  we  have  6(^  =  a6.  ^^^-  ^ 

With  0  as  origin  and  OAq  as  axis  of  x  we  have 

X  =  (a -\- b)  cos  <j> -i- a  sin  [^  —  (i  tt  —  <^)], 
y  :=  {a -\- b)  sin  (f>  —  a  cos  [^  —  (|-  tt  —  <^)], 

I.e.  X  =  (a -\- b)  cos  (f>  —  a  cos 9, 


y=:  (a  -\-  b)  sin  <f)  —  a 


sin 


a 

a  +  b 
a 


<t>- 


174  PLANE  ANALYTIC  GEOMETRY        [IX,  §  182 

182.  H3rpocycloid.  If  the  circle  rolls  on  the  inside  of  the 
fixed  circle,  the  path  of  any  point  of  the  rolling  circle  is  called 
a  hypocycloid.  The  equations  are  obtained  in  the  same  way ; 
they  differ  from  those  of  the  epicycloid  (§  181)  merely  in  hav- 
ing a  replaced  by  —  a.     Write  down  these  equations. 

Show  that :  (a)  for  b  =  2a  the  hypocycloid  reduces  to  a 
straight  line,  and  illustrate  this  graphically ;  (b)  for  6  =  4  a  the 
curve,  called  the  four-cusped  hypocycloid,  has  the  equations 

x  =  Sa  cos  <f>-\-a  cos  3  <^  =  a  cos'  <^, 
y=Sa  sin  <f>  —  asmS<f>  =  a sin'  </>, 

whence  x^  -\-y^  =  a^, 

EXERCISES 

1.  Sketch  the  following  curves :  (a)  Spiral  of  Archimedes  r  =  ai>; 
(6)  Hyperbolic  spiral  r<f>  =  a  ;  (c)  Lituus  r'^tp  =  a^. 

2.  Sketch  the  following  curves  :  (a)  r  =  a  sin  </> ;  (6)  r  =  a  cos  0  ; 
(c)  r  =  a8in2  0  ;  (d)  r  =  acos2<f>;  (e)  r=  acos30  ;  (/)  r  =  asin30; 
(g)  r  =  a  cos  4  0 ;  {h)  r  =  a  sin  4  <f>. 

3.  Sketch  with  respect  to  the  same  axes  the  Cassinian  ovals  (§  178) 
for  a  =  1  and  k  =  2,  1.5,  1.1,  1,  .75,  .5,  0. 

4.  Let  two  perpendicular  lines  AB  and  CD  intersect  at  O.  Through 
a  fixed  point  Q  of  AB  draw  any  line  intersecting  CD  at  R.  On  this  line 
lay  off  in  both  directions  from  B  segments  BP  of  length  OB.  The  locus 
of  P  is  called  the  strophoid.     Find  the  equation  and  sketch  the  curve. 

6.  Show  that  the  lemniscate  (§  178)  is  the  inverse  curve  of  an  equi- 
lateral hyperbola  with  respect  to  a  circle  about  its  center. 

6.  Show  that  the  strophoid  (Ex.  4)  is  the  curve  inverse  to  an  equilat- 
eral hyi)erbola  with  respect  to  a  circle  about  a  vertex  with  radius  equal 
to  the  transverse  axis. 

7.  Show  that  the  cissoid  (§  176)  is  the  curve  inverse  to  a  parabola 
with  respect  to  a  circle  about  its  vertex. 

8.  Find  the  curve  inverse  to  the  cardioid  (§175)  with  respect  to  a 
circle  about  the  origin. 


IX,  §  183] 


SPECIAL  CURVES 


175 


9.   Transform  the  equation  a{x^  4-  y'^)  =  y?  to  polar  coordinates,  in- 
dicate a  geometrical  construction,  and  draw  the  curve. 

10.  A  tangent  to  a  circle  of  radius  2  a  about  the  origin  intersects  the 
axes  at  T  and  T.     Find  and  sketch  the  locus  of  the  midpoint  P  of  TT'. 

11.  From  any  point  ^  of  the  line  x  —  a  draw  a  line  parallel  to  the  axis 
Ox  intersecting  the  axis  Oy  at  Q.  Find  and  sketch  the  locus  of  the  foot 
of  the  perpendicular  from  (7  on  0§. 

12.  The  center  of  a  circle  of  radius  a  moves  along  the  axis  Ox.  Find 
and  sketch  the  locas  of  the  intersections  of  this  circle  with  lines  joining 
the  origin  to  its  highest  point. 

13.  The  center  of  a  circle  of  radius  a  moves  along  the  axis  Ox.  Find 
and  sketch  the  locus  of  its  points  of  contact  with  the  lines  through  the  origin. 

183.  The  Sine  Curve.  The  simple  sine  curvCj  y=8m  x, 
is  best  constructed  by  means  of  an  auxiliary  circle  of  radius 
one.  In  Fig.  100,  OQ  is  made  equal  to  the  length  of  the  arc 
OA  =  X ;  the  ordinate  at  Q  is  then  equal  to  the  ordinate  BA  of 
the  circle. 

y 


Fig.  100 

Construct  one  whole  period  of  the  sine  curve,  i.e.  the  portion 
corresponding  to  the  whole  circumference  of  the  auxiliary 
circle ;  the  width  2  tt  of  this  portion  is  called  the  period  of  the 
function  sin  x. 

The  simple  cosine  curve,  y  =  GOSx,  is  the  same  as  the  sine 
curve  except  that  the  origin  is  taken  at  the  point  (Jir,  0). 

The  simple  tangent  curve,  y  =  tan  x,  is  derived  like  the  sine 
curve  from  a  unit  circle.    Its  period  is  tt. 


176 


PLANE  ANALYTIC  GEOMETRY        [IX,  §  184 


184.  The  Inverse  Trigonometric  Curves.  The  equation 
y  =  sin  X  can  also  be  written  in  the  form 

X  =  sin~^  y,     ox  x  —  arc  sin  y. 

The  curve  represented  by  this  equation  is  of  course  the  same 
as  that  represented  by  the  equation  y  =  sin  x. 

But  if  X  and  y  be  interchanged,  the  resulting  equation 

x  =  sin  y,  OT  y  =  sin"*  ^,  y  =  arc  sin  x, 

represents  the  curve  obtained  from  the  simple  sine  curve  by 
reflection  in  the  line  y=x(^  70). 

Notice  that  the  trigonometric  functions  sina;,  cos  x,  tan  x,  etc., 
are  one-valued,  i.e.  to  every  value  of  x  belongs  only  one  value 
of  the  function,  while  the  inverse  trigonometric  functions  sin~*  x, 
COS"* a;,  tan"*  a;,  etc.,  are  many-valued;  indeed,  to  every  value  of 
X,  at  least  in  a  certain  interval,  belongs  an  infinite  number  of 
values  of  the  function. 


185.   Transcendental  Curves.     The  trigonometric  and  in- 
verse trigonometric  curves,  as  well  as,  in  general,  the  cycloids 

and  trochoids,  are  transcen- 
dental curves,  so  called  because 
the  relation  between  the  carte- 
sian coordinates  x,  y  cannot  be 
expressed  in  finite  form  (i.e. 
without  using  infinite  series)  by 
means  of  the  algebraic  opera- 
tions of  addition,  subtraction, 
multiplication,  division,  and 
raising  to  a  power  with  a  con- 
Fio.  lOX  stant  exponent. 


IX,  §186]  SPECIAL   CURVES  177 

186.   Logarithmic  and  Exponential  Curves.     Another  very 
important  transcendental  curve  is  the  exponential  curve 

y  =  a% 
and  its  inverse,  the  logarithmic  curve 

y  =  loga  X, 

where  a  is  any  positive  constant  (Fig.  101).    A  full  discussion 
of  these  curves  can  only  be  given  in  the  calculus. 

EXERCISES 

1.  From  a  table  of  trigonometric  functions,  plot  the  curve  y  =  sinoj. 

2.  Plot  the  curve  y  =  sinx  geometrically,  as  in  §  183. 

3.  Plot  the  curve  y  =  cos  x  (a)  from  a  table ;  (6)  by  a  geometric  con- 
struction similar  to  that  of  §  183. 

4.  Plot  the  curve  y  =  tanaj  from  a  table. 

5.  Plot  each  of  the  curves 

(a)  y  =  sm2  x.  (6)  y  =  2  cos  Sx.  (c)  y  =  S  tan  (x/2). 

(df)  y  =  sec  x.  (e)  y  =  cot  2  sc.  (f)y  =  2  tan  4  x. 

6.  Plot  each  of  the  curves 

(a)  y  =  sin-i  X.  (6)  y  =  cos~i  x.  (c)  y  =  tan-i  x, 

7.  By  adding  the  ordinates  of  the  two  curves  y  =  s'mx  and  y  =  cos  x, 
construct  the  graph  of  y  =  sin  x  +  cos  x. 

8.  Draw  each  of  the  curves 

(a)  y  =  smx  +  2  cos  x.  (c)  y  =  sec  x  +  tan  x. 

(6)  ?/  =  2  sin  a;  +  cos(x/2).  (d)  2/  =  sin  x  -f  2  sin  2  a;  +  3  sin  3x. 

9.  The  equation  aj  =  sin  t,  where  t  means  the  time  and  x  means  the 
distance  of  a  body  from  its  central  position,  represents  a  Simple  Harmonic 
Motion.     From  the  graph,  describe  the  nature  of  the  motion. 

10.  From  a  table  of  logarithms  of  numbers,  draw  the  curve  y—\ogiox. 

11.  By  multiplying  the  ordinates  of  the  curve  of  Ex.  10  by  3,  construct 
the  curve  y  =  S  logio  x. 

12.  From  the  figure  of  Ex.  10,  construct  the  curve  y  =  10*  by  reflec- 
tion of  the  curve  of  Ex.  10  in  the  line  y  =  x. 

13.  Draw  the  curve  y  =  ^  logio^:  by  the  process  of  Ex.  11.     Show  that 
it  represents  the  equation  y  =  logiooaJ,  since 

y  =  logioo  X  =  logioo  10  X  logio  x  =  l  logio  x. 

N 


178  PLANE  ANALYTIC  GEOMETRY        [IX,  §  187 

PART   III.     EMPIRICAL  EQUATIONS 

187.  Empirical  Formulas.  In  scientific  studies,  the  rela- 
tions between  quantities  are  usually  not  known  in  advance, 
but  are  to  be  found,  if  possible,  from  pairs  of  numerical  values 
of  the  quantities  discovered  by  experiment. 

Simple  cases  of  this  kind  have  already  been  given  in  §§  15, 
29.  In  particular,  the  values  of  a  and  b  in  formulas  of  the 
type  y  =  a-\-hx  were  found  from  two  pairs  of  values  of  x  and  y. 
Compare  also  §  34. 

Likewise,  if  two  quantities  y  and  x  are  known  to  be  connected 
by  a  relation  of  the  form  ?/  =  a  -f  6a;  +  cx^,  the  values  of  a,  6,  c 
can  be  found  from  any  three  pairs  of  values  of  x  and  y.  For, 
if  any  pair  of  values  of  x  and  y  are  substituted  for  x  and  y 
in  this  equation,  we  obtain  a  linear  equation  for  a,  6,  and  c. 
Three  such  equations  usually  determine  a,  6,  and  c. 

In  general  the  coefficients  a,  h,  c,  •••,  Z  in  an  equation  of  the 

^^®  y:=a-\-hx-\-cy?+  ...  -\-lxr 

can  be  found  from  any  n  +  1  pairs  of  values  of  x  and  y. 

188.  Approximate  Nature  of  Results.  Since  the  measure- 
ments made  in  any  experiment  are  liable  to  at  least  small 
errors,  it  is  not  to  be  expected  that  the  calculated  values  of 
such  coefficients  as  a,  6,  c,  .-.  of  §  187  will  be  absolutely  accu- 
rate, nor  that  the  points  that  represent  the  pairs  of  values  of 
X  and  y  will  all  lie  absolutely  on  the  curve  represented  by  the 
final  formula. 

To  increase  the  accuracy,  a  large  number  of  pairs  of  values 
of  X  and  y  are  usually  measured  experimentally,  and  various 
pairs  are  used  to  determine  such  constants  as  a,b,Cf  .••  of  §  187. 
The  average  of  all  the  computed  values  of  any  one  such  con- 
stant is  often  taken  as  a  fair  approximation  to  its  true  value. 


IX,  §  189] 


EMPIRICAL  EQUATIONS 


179 


189.   Illustrative  Examples. 

Example  1.     A  wire  under  tension  is  found  by  experiment  to  stretch 
an  amount  I,  in  thousandths  of  an  inch,  under  a  tension  T,  in  pounds,  as 

follows :  — 

T  in  pounds  . " 10  15  20  25  30 

I  in  thousandths  of  an  inch  .        8  12.5         15.5  20  23 

Find  a  relation  of  the  form  I  =  kT  (Hookers  Law)  which  approx- 
imately represents  these  results. 

First  plot  the  given  data  on  squared  paper,  as  in  the  adjoining  figure. 
30 


25 


20 


15 


10 


ZZIZZZy^LZZZZZZZZIZZZZZ 


10 


15 


20 


25 


30 


35 


Fig.  102 


Substituting  I  =  8,  T  =  10  in  I  =  kT,  we  find  k  =  .8.  From  I  =  12.5, 
T  =  15,  we  find  k  =  .833.  Likewise,  the  other  pairs  of  values  of  I  and  T 
give,  respectively,  k  =  .775,  k  =  .S,  k=  .767.  The  average  of  all  these 
values  of  A;  is  A;  =  .795  ;  hence  we  may  write,  approximately, 

I  =  .795  T. 


180 


PLANE  ANALYTIC  GEOMETRY        [IX,  §  189 


This  equation  is  represented  by  the  line  in  Fig.  102 ;  this  line  does  not 
pass  through  even  one  of  the  given  points,  but  it  is  a  fair  compromise  be- 
tween all  of  them,  in  view  of  the  fact  that  each  of  them  is  itself  probably 
slightly  inaccurate. 

Example  2.  In  an  experiment  with  a  Weston  Differential  Pulley 
Block,  the  effort  E^  in  pounds,  required  to  raise  a  load  IF,  in  pounds,  was 
found  to  be  as  follows  : 


w 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

E 

3i 

4| 

6i 

n 

9 

lOi 

]2i 

13| 

16 

16i 

Find  a  relation  of  the  form  E  =  aW -\-  h  that  approximately  agrees 
with  these  data.  [Gibson] 

These  values  may  be  plotted  in  the  usual  manner  on  squared  paper. 
They  will  be  found  to  lie  very 


20 


10 


nearly  on  a  straight  line.  If  E 
is  plotted  vertically,  6  is  the  in- 
tercept on  the  vertical  axis,  and 
a  is  the  slope  of  the  line  ;  both 
can  be  measured  directly  in  the 
figure. 

To  determine  a  and  h  more 
exactly,  we  may  take  various 
points  that  lie  nearly  on  the 
line.  Thus  {E  =  Q\,  11'=  30) 
and  {E  =  16J,  W  =  100)  lie 
nearly  on  a  line  that  passes  close 
to  all  the  points.    Substituting  in  the  equation  E 


m 


JW 


20 


40 
Fig. 


GO 

103 


80 


100. 


6  J  =  30  a  +  6,       16J 


aW-hhwe  obtain 
100  a  +  6, 


whence  a  =  0.146,  6  =  1. 


!6.     Hence  we  may  take 
^=0.146  TF+  1.86, 


approximately.  Other  pairs  of  values  of  E  and  W  may  be  used  in  like 
manner  to  find  values  for  a  and  b,  and  all  the  values  of  each  quantity  may 
be  averaged. 


50 

60 

70 

80 

90 

205° 

226° 

250°  ' 

'  276° 

304* 

IX,  §  189]  EMPIRICAL  EQUATIONS  181 

Example  3.  If  6  denotes  the  melting  point  (Centigrade)  of  an  alloy 
of  lead  and  zinc  containing  x  per  cent  of  lead,  it  is  found  that 

X  =  %  lead 40 

e  =  melting  point     ....      186° 

Find  a  relation  of  the  form  6  =  a  -{-  bx  -\-  cx^  that  approximately  expresses 
these  facts.  [Saxelby] 

Taking  any  three  pairs  of  values,  say  (40,  186),  (70,  250),  (90,  304), 
and  substituting  in  d  =  a  +  bx  +  cx^  we  find 

186  =  a  +  40  &  4-  1600  c, 
260  =  a +  70  b  +  4900  c, 
304  =  a  +  90  &  +  8100  c, 

whence  a  =  132,  b  =  .92,  c  =  .0011,  approximately ;  whence 
^  =  132  +  .92  X  +  .0011  x^. 
Other  sets  of  three  pairs  of  values  of  x  and  y  may  be  used  in  a  similar 
manner  to  determine  a,  &,  c  ;  and  the  resulting  values  averaged,  as  above. 

EXERCISES 

1.  In  experiments  on  an  iron  rod,  the  amount  of  elongation  I  (in  thou- 
sandths of  an  inch)  and  the  stretching  force  p  (in  thousands  of  pounds) 
were  found  to  be  {p  =  10,  I  =8 ),  (p  =  20,  I  =  15),  (p  =  40,1  =  31). 
Find  a  formula  of  the  type  I  =  Jc-p  which  approximately  expresses  these 
data.  Ans.     k  =  .715. 

2.  The  values  1  in.  =2,5  cm.  and  1  ft.  =  30.5  cm.  are  frequently 
quoted,  but  they  do  not  agree  precisely.  The  number  of  centimeters,  c, 
in  i  inches  is  surely  given  by  a  formula  of  the  type  c  =  ki.  Find  k  ap- 
proximately from  the  preceding  data. 

3.  The  readings  of  a  standard  gas-meter  S  and  those  of  a  meter  T  being 
tested  on  the  same  pipe-line  were  found  to  be  (aS^=3000,  T'=0),  (/S'=3510, 
T  =  500),  {S  =  4022,  T  =  1000).  Find  a  formula  of  the  type  T  =  aS+b 
which  approximately  represents  these  data. 

4.  An  alloy  of  tin  and  lead  containing  x  per  cent  of  lead  melts  at  the 
temperature  9  (Fahrenheit)  given  by  the  values  (a!  =  25%,  ^  =  482°), 
(x  =  50  %,  d  =  370°),  (x  =  75  %,  6  =  356°) .  Determine  a  formula  of  the 
type  d  =  a  +  bx  +  cx'^  which  approximately  represents  these  values. 


182  PLANE  ANALYTIC  GEOMETRY        [IX,  §  189 

5.  The  temperatures  0  (Centigrade)  at  a  depth  d  (feet)  below  the  sur- 
face of  the  earth  in  a  mine  were  found  to  be  d  =  100,  d  =  15.7°  ;  d  =  200, 
^=16.5  ;  J =300,  ^=17.4.  Find  a  relation  of  the  form  e=a-\-bd  between 
d  and  d. 

6.  Determine  a  line  that  passes  reasonably  near  each  of  the  three 
points  (2,  4),  (C,  7),  (10,  9).  Determine  a  quadratic  expression 
y=a-\-bx-\-cx^  that  represents  a  parabola  through  the  same  three  points. 

7.  Determine  a  parabola  whose  equation  is  of  the  form  y=a+bx+cx^ 
that  passes  through  each  of  the  points  (0,  2.5;,  (1.5,  1.5),  and  (3.0,  2.8). 
Are  the  values  of  rt,  &,  c  changed  materially  if  the  point  (2.0,  1.7)  is 
substituted  for  the  point  (1.5,  1,5)  ? 

8.  If  the  curve  y  =  sin  a;  is  drawn  with  one  unit  space  on  the  x-axis 
representing  60^,  the  points  (0,  0),  (^,  J),  (1|,  1)  lie  on  the  curve.  Find  a 
parabola  of  the  form  y  =  a+bx-\-cx^  through  these  three  points,  and  draw 
the  two  curves  on  the  same  sheet  of  paper  to  compare  them. 

190.  Substitutions.  It  is  particularly  easy  to  test  whether 
points  that  are  given  by  an  experiment  really  lie  on  a  straight 
line ;  that  is,  whether  the  quantities  measured  satisfy  an  equa- 
tion of  the  form  y  =  a-{-bx.  This  is  done  by  means  of  a  trans- 
parent ruler  or  a  stretched  rubber  band. 

For  this  reason,  if  it  is  suspected  that  two  quantities  x  and 
y  satisfy  an  equation  of  the  form 

it  is  advantageous  to  substitute  a  new  letter,  say  u,  for  a? : 

u  =  x^,      y  =  a-{-hu 
and  then  plot  the  values  of  y  and  u.     If  the  new  figure  does 
agree  reasonably  well  with  some  straight  line,  it  is  easy  to  find 
a  and  &,  as  in  §  189. 

Likewise,  if  it  is  suspected  that  two  quantities  x  and  y  are 
connected  by  a  relation  of  the  form 

y  =  a-\-h--OTxy  =  ax-\-hf 

X 

it  is  advantageous  to  make  the  substitution  u  =  1/x. 


IX,  §  191]  EMPIRICAL  EQUATIONS  183 

Other  substitutions  of  the  same  general  nature  are  often 
useful. 

In  any  case,  the  given  values  of  x  and  y  should  be  plotted  first 
unchanged,  in  order  to  see  what  substitution  might  be  useful. 

191.  Illustrative  Example.  If  a  body  slides  down  an  inclined 
plane,  the  distance  s  that  it  moves  is  connected  with  the  time  t  after  it 
starts  by  an  equation  of  the  form  s  =  kt'^.  Find  a  value  of  k  that  agrees 
reasonably  with  the  following  data : 

s,  in  feet 2.6  10.1  23.0  40.8  63.7 

t,  in  seconds 1  2  3  4  5 

In  this  case,  it  is  not  necessary  to  plot  the  values  of  s  and  t  themselves, 
because  the  nature  of  the  equation,  s  =  kt^,  is  known  from  physics. 

Hence  we  make  the  substitution  t^  =  u,  and  write  down  the  supple- 
mentary table : 

8,  in  feet 2.6  10.1  23.0         40.8         63.7 

w  (or  t2) 1  4  9  16  25 

These  values  will  be  found  to  give  points  very  nearly  on  a  straight  line 
whose  equation  is  of  the  form  s  —  ku.  To  find  A;,  we  divide  each  value  of 
s  by  the  corresponding  value  of  u  ;  this  gives  several  values  of  k : 

k  2.6        2.525        2.556         2.55         2.548 

The  average  of  these  values  of  k  is  approximately  2.556  ;  hence  we  may 
write  s  =  2.556  M,  or  s  =  2.556  t^. 

EXERCISES 

1.  Find  a  formula  of  the  type  u  =  kv^  that  represents  approximately 
the  following  values : 


u 

3.9 

15.1 

34.5 

61.2 

95.5 

137.7 

187.4 

V 

1 

2 

3 

4 

5 

6 

7 

184  PLANE  ANALYTIC  GEOMETRY        [IX,  §  191 

2.  A  body  starts  from  rest  and  moves  s  feet  in  t  seconds  according  to 
the  following  measured  values  : 

s,  in  feet 3.1         13.0        30.6        50.1        79.5       116.4 

t,  in  seconds 5  1  1.5  2  2.5  3 

Find  approximately  the  relation  between  s  and  t. 

3.  The  pressure  p,  measured  in  centimeters  of  mercury,  and  the  volume 
V,  measured  in  cubic  centimeters,  of  a  gas  kept  at  constant  temperature, 
were  found  to  be  : 

191 


V 

146 

155 

165 

178 

p 

117.2 

109.4 

102.4 

95.0 

Substitute  u  for  1/v,  compute  the  values  of  m,  and  determine  a  relation 
of  the  form  p  =  ku;  that  is,  p  =  k/v. 

4.   Determine  a  relation  of  the  form  y  =  a  +  bx^  that  approximately 
represents  the  values : 

X  1  2  3  4  5  6  7 

y  14.1  25.2  44.7         71.4         105.6        147.9        197.7 

192.   Logarithmic  Plotting.      In  case  the  quantities  y  and  x 
are  connected  by  a  relation  of  the  form 

it  is  advantageous  to  take  logarithms  (to  the  base  10)  on  both 

sides : 

log  y  =  log  kaf"  =  log  k  -\-n  log  x, 

and  then  substitute  new  letters  for  log  x  and  log  y : 

u  =  log  X,        v  =  log  y. 

For,  if  we  do  so,  the  equation  becomes 

v  =  l  -\-  nuj 
where  I  =  log  k. 


IX,  §  192] 


EMPIRICAL  EQUATIONS 


185 


If  the  values  of  x  and  ?/  are  given  by  an  experiment,  and  if 
w  =  log  £c  and  v  =  log  y  are  computed,  the  values  of  u  and  v 
should  correspond  to  points  that  lie  on  a  straight  line,  and  the 
values  of  I  and  n  can  be  found  as  in  §  189.  The  value  of  k 
may  be  found  from  that  of  Z,  since  log  k=l. 

Example  1.  The  amount  of  water  J.,  in  cu.  ft.,  that  will  flow  per 
minute  through  100  feet  of  pipe  of  diameter  d,  in  inches,  with  an  initial 
pressure  of  50  lb.  per  sq.  in. ,  is  as  follows  : 


1 


1.5 
13.4.3 


2 
27.50 


3 
75.13 


4 

152.51 


6 
409.54 


Find  a  relation  between  A  and  d. 


Let  u  =  \ogd,  V  =  log  A  ;  then  the  values  of  u  and  v  are 


\ogd  . 
losA  . 


0.000 
0.688 


0.176 
1.128 


0.301 
1.439 


0.477 
1.876 


0.602 
2.183 


0.778 
2.612 


"  J  .2  .3  4  .5  .6  .7         .8 

Fig.  104 

These  values  give  points  in  the  (w,  v)  plane  that  are  very  nearly  on 
a  straight  line  ;  hence  we  may  write,  approximately, 

where  a  and  b  can  be  determined  directly  by  measurement  in  the  figure, 


186  PLANE  ANALYTIC  GEOMETRY        [IX,  §  192 

or  as  in  §  189.     If  we  take  the  first  and  last  pairs  of  values  of  u  and  v,  we 

find 

.688  =  a +  0, 

2.612  =  a +.778  ft. 

Solving  these  equations,  we  find  approximately,  a  =  .688,  b  =  2.473, 
and  we  may  write 

V  =  .688  +  2.473  u    or    log  A  =  .688  +  2.473  log  d. 

Since  .688  =  log  4.88, 

the  last  equation  may  be  written  in  the  form 

log  ^  =  log  4. 88  +  2. 473  log  d 

=  log(4.88  (P««) 

whence  ^  =  4.88  ^2.478. 

Slightly  different  values  of  the  constants  may  be  found  by  using  other 
pairs  of  values  of  u  and  v. 

193.  Logarithmic  Paper.  Paper  called  logarithmic  paper 
may  be  bought  that  is  ruled  in  lines  whose  distances,  horizon- 
tally and  vertically,  from  one  point  0  (Fig.  105)  are  propor- 
tional to  the  logarithms  of  the  numbers  1,  2,  3,  etc. 

Such  paper  may  be  used  advantageously  instead  of  actually 
looking  up  the  logarithms  in  a  table,  as  was  done  in  §  192. 
For  if  the  given  values  be  plotted  on  this  new  paper,  the  result- 
ing figure  is  identically  the  same  as  that  obtained  by  plotting 
the  logarithms  of  the  given  values  on  ordinary  squared  paper. 

Example.  A  strong  rubber  band  stretched  under  a  pull  of  p  kg. 
shows  an  elongation  of  E  cm.  The  following  values  were  found  in  an  ex- 
periment : 

p        0.5      1.0     1.5     2.0     2.5     3.0     3.5     4.0     4.5     5.0     6.0     7.0 
E        0.1      0.3     0.6     0.9     1.3     1.7     2.2     2.7     3.3     3.9     5.3     6.9 

[RiGGS] 

If  these  values  are  plotted  on  logarithmic  paper  as  in  Fig.  105,  it  is  evi- 
dent that  they  lie  reasonably  near  a  straight  line,  such  as  that  drawn. 


IX,  §  193] 


EMPIRICAL  EQUATIONS 


187 


By  measurement  in  the  figure,  the  slope  of  this  line  is  found  to  be  1.6, 
approximately.     Hence  if  u  =  logp  and  v  =  log  E,  we  have 

where  Z  is  a  constant  not  yet  determined ;  whence 

log  E=l  +  1.6logp 
or  E  =  kp^-^ 


l(p 

.IJ  -  -  ■  ■ 

7 

Q 

/ 

/ 

^  r 

_J 

/ 

/ 

/ 

■   j&  =  elongation  in  c 
p  —  pull  in  kg. 

/ 

m. 

/ 

/ 

Z  z-zz 

TTfTTI — I — '"'"""^ 

/. 





— ^ 

t-: 

EE 

15 

— " 

J 

/ 

■  - _  s  i. 

/ 

i 

it 

'fi 

^/  -    — 

fe  _    

/ 

/ 

~t 

2 

*i  _  __ .. 

^'-^.Z  J 

^  ----- 

A 

4=  =  =  :: 

Z'l^'^. 

■ 



_^ 

EE 

45 

2 



— 

.■ 

^          -I 

"-I2 

1 

:^ 

1 

y 

--  P 

^4        . 

15     .2 

.2 

i     ,4 

i,j 

i  .7.ajSi 

16     2 

'  2 

I       4 

5    € 

»  7  8 

9i0 

Fig.  105.  —  Elongation  of  a  Rubber  Band 

where  Z  =  log  A:.     If  p  =  1,  ^  =  A: ;  from  the  figure,  if  p  =  1,  E  =  .B 
hence  A:  =  .3,  and 

E  =  .3i>i-6. 

The  use  of  logarithmic  paper  is  however  not  at  all  essential ; 
the  same  results  may  b©  obtained  by  the  method  of  §  192. 


188  PLANE  ANALYTIC  GEOMETRY         [IX,  §  193 

EXERCISES 

1.  In  testing  a  gas  engine  corresponding  values  of  the  pressure  p,  meas- 
ured in  pounds  per  square  foot,  and  the  volume  r,  in  cubic  feet,  were 
obtained  as  follows  :  tJ  =  7.14,  p  =  54.6  ;  7.73,  60.7  ;  8.59,  45.9.  Find 
the  relation  between |)  and  v  (use  logarithmic  plotting). 

Ans.  p  =  387.6  v-^,  or  pv^  =  387.6. 

2.  Expansion  or  contraction  of  a  gas  is  said  to  be  adiabatic  when  no 

heat  escapes  or  enters.   Determine  the  adiabatic  relation  between  pressure 

p  and  volume  v  (Ex.   1)  for  air  from  the   following  observed  values: 

p  =  20.64,  V  =  Q.'27  ;  26.79,  6.34 ;  64.25,  3.15. 

Ans.  pri«  =  273.5. 

3.  The  intercollegiate  track  records  for  foot-races  are  as  follows, 
where  d  means  the  distance  run,  and  t  means  the  record  time  : 

d  100  yd.      220  yd.      440  yd.      880  yd.        1  mi.  2  mi. 

t  0:09^  0:21^  0:48  1:54|  4:15§  9:24^ 

Plot  the  logarithms  of  these  values  on  squared  paper  (or  plot  the 
given  values  themselves  on  logarithmic  paper).      Find  a  relation  of  the 
form  t  =  kd^.     What  should  be  the  record  time  for  a  race  of  1320  yd.  ? 
[See  Kennelly,  Popular  Science  Monthly,  Nov.  1908.] 

4.  Solve  the  Example  of  §  193  by  the  method  of  §  192. 

6.  Each  of  the  following  sets  of  quantities  was  found  by  experiment. 
Find  in  each  case  an  equation  connecting  the  two  quantities,  by  §§  192- 
193. 


(«)    V 

p 

1 

137.4 

2 
62.6 

3 
39.6 

4 
28.6 

5 

22.6 

(b)  u 

V 

12.9 
63.0 

17.1 
27.0 

23.1 
13.8 

28.6 
8.6 

3.0 
6.9 

(c)  0 
c 

82° 
2.09 

212° 
2.69 

390° 
2.90 

570° 
2.98 

750° 
3.09 

1100° 
3.28 

SOLID    ANALYTIC    GEOMETRY 


CHAPTER   X 


COORDINATES 


194.  Location  of  a  Point.  The  position  of  a  point  in  three- 
dimensional  space  can  be  assigned  without  ambiguity  by  giv- 
ing its  distances  from  three  mutually  rectangular  planes,  pro- 
vided these  distances  are  taken  with  proper  signs  according  as 
the  point  lies  on  one  or  the  other  side  of  each  plane. 

The  three  planes,  each  perpendicular  to  the  other  two,  are 
called  the  coordinate  planes ;  their  common  point  0  (Fig.  106) 
is  called  the  origin.    The  three  ^ 

mutually  rectangular  lines  Ox,  q^ ^pf 

Oy,  Oz  in  which  the  planes  in- 
tersect are  called  the  axes  of 
coordinates;  on  each  of  them 
a  positive  sense  is  selected 
arbitrarily,  by  affixing  the 
letter  x,  y,  z,  respectively,  y^ 

The  three  coordinate  planes, 
Oyz,  Ozx,  Oxy,  divide  the  whole 
of  space  into  eight  compartments  called  octants.  The  first 
octant  in  which  all  three  coordinates  are  positive  is  also  called 
the  coordinate  trihedral. 

If  P',  P",  P"'  are  the  projections  of  any  point  P  on  the 
coordinate  planes  Oyz,  Ozx,  Oxy,  respectively,  then  P'P  =  x, 
P"P  =  y,  P"'P=z  are  the  rectangular  cartesian  coordinates  of 

189 


?<- 


^Q" 


Fig.  106 


190 


SOLID  ANALYTIC  GEOMETRY         [X,  §  194 


P.  If  the  planes  through  P  parallel  to  Oyz,  Ozx,  Oxy  intersect 
the  axes  Ox,  Oy,  Oz  in  Q\  Q",  Q'",  the  point  P  is  found  from 
its  coordinates  x,  y,  z  by.  passing  along  the  axis  Ox  through  the 
distance  OQ'=x,  parallel  to  Oy  through  the  distance  Q'P"=y, 
and  parallel  to  Oz  through  the  distance  P"P=z,  each  of 
these  distances  being  taken  with  the  proper  sense. 

Every  point  in  space  has  three  definite  real  numbers  as  coordi- 
nates; conversely,  to  every  set  of  three  real  numbers  corresponds 
one  and  only  one  point. 

Locate  the  points :  (2,  3,  4),  (-  3,  2,  0),  (5,  0,  -  3),  (0,  0,  4), 
(0,  -  6,  0),  (-  5,  -  8,  -2). 

195.  Distance  of  a  Point  from  the  Origin.  For  the  distance 
OP=r  (Fig.  106)  of  the  point  P(x,  y,  z)  from  the  origin  0  we 
have,  since  OP  is  the  diagonal  of  a  rectangular  parallelepiped 
with  edges  OQ'  =  x,  OQ"  =  y,  OQ"'  =  z: 


r=  Vic^  -f  2/^  + 

196.  Distance  between  two  Points, 
the  t>vo  points  Pi  (x^ ,  ?/i ,  z^)  and  Pj 
(X2,y2, Z2)  can  be  found  if  the  coordi- 
nates of  the  two  points  are  given. 
For  (Fig.  107),  the  planes  through  P^ 
and  those  through  Pj  parallel  to  the 
coordinate  planes  bound  a  rectangular 
parallelepiped  with  P1P2  =  d  as  di- 
agonal ;  and  as  its  edges  are 

P^Q=zX2-x,,    P,R  =  7j2-yi., 
we  find 


The  distance  between 


M 


<?^ 


^ 


Fig.  107 

PlS  =  Z2  —  Zi, 


d  =  V(a^  -  x,y  -h  (t/2  -  y,y  +  (z,  -  Zy)\ 

197.  Oblique  Axes.  The  position  of  a  point  P  in  space  can  also 
be  determined  with  respect  to  three  axes  not  at  right  angles.  The  coor- 
dinates of  P  are  the  segments  cut  o2  on  the  axes  by  planes  through  P 


X,  §  197]  COORDINATES  191 

parallel  to  the  coordinate  planes.    In  what  follows,  the  axes  are  always 
assumed  to  he  at  right  angles  unless  the  contrary  is  definitely  stated. 

EXERCISES 

1.  What  are  the  coordinates  of  the  origin  ?  What  can  you  say  of  the 
coordinates  of  a  point  on  the  axis  Ox  ?  on  the  axis  Oy  ?  on  the  axis  Oz  ? 

2.  What  can  you  say  of  the  coordinates  of  a  point  that  lies  in  the 
plane  Oxy  ?  in  the  plane  Oyz  ?  in  the  plane  Ozx  ? 

3.  Where  is  a  point  situated  when  x  =  0?  when  z  =  0?  when 
x  =  y  =  0?  when  y  =  z?  when  x  =  2?  when  z  =—Z?  when  x  =  1 , 
y  =  2? 

4.  A  rectangular  parallelepiped  lies  in  the  first  octant  with  three  of 
its  faces  in  the  coordinate  planes,  its  edges  are  of  length  a,  &,  c,  respec- 
tively ;  what  are  the  coordinates  of  the  vertices  ? 

6.  Show  that  the  points  (4,3,  5),  (2,  -1,3),  (0,1,7)  are  the 
vertices  of  an  equilateral  triangle. 

6.  Show  that  the  points  (-  1,  1,  3),  (—  2,  —  1,  4),  (0,  0,  5)  lie  on  a 
sphere  whose  center  is  (2,  —  3,  1).     What  is  the  radius  of  this  sphere  ? 

7.  Show  that  the  points  (6,  2,  -  5),  (2,  -  4,  7),  (4,  -  1,  1)  lie  on  a 
straight  line. 

8.  Show  that  the  triangle  whose  vertices  are  (a,  6,  c) ,  (&,  c,  a) ,  (c,  a,  &) 
is  equilateral. 

9.  What  are  the  coordinates  of  the  projections  of  the  point  (6,  3,  —  8) 
on  the  axes  of  coordinates  ?  What  are  the  distances  of  this  point  from  the 
coordinate  axes  ? 

10.  What  is  the  length  of  the  segment  of  a  line  whose  projections  on 
the  coordinate  axes  are  5,  3,  and  2  ? 

11.  What  are  the  coordinates  of  the  points  which  are  symmetric  to 
the  point  (a,  6,  c)  with  respect  to  the  coordinate  planes  ?  with  respect 
to  the  axes  ?  with  respect  to  the  origin  ? 

12.  Show  that  the  sum  of  the  squares  of  the  four  diagonals  of  a  rec- 
tangular parallelepiped  is  equal  to  the  sum  of  the  squares  of  its  edges. 


192  SOLID  ANALYTIC  GEOMETRY  [X,  §  198 

198.  Projection.  The  projection  of  a  point  on  a  plane  or 
line  is  the  foot  of  the  perpendicular  let  fall  from  the  point  on 
the  plane  or  line.  The  projection  of  a  rectilinear  segment  AB 
on  a  plane  or  line  is  the  intercept  A'B'  between  the  feet  of  the 
perpendiculars  AA',  BB'  let  fall  from  Ay  B  on  the  plane  or 
line.  If  a  is  one  of  the  two  angles  made  by  the  segment  with 
the  plane  or  line,  we  have 

A'B'  =  AB  cos  a. 

In  analytic  geometry  we  have  generally  to  project  a  vector, 
i.e.  a  segment  with  a  definite  sense,  on  an  axis,  i.e.  on  a  line 
with  a  definite  sense  (compare  §  19).  The  angle  a  is  then 
understood  to  be  the  angle  between  the  positive  senses  of 
vector  and  axis  (both  being  drawn  from  a  common  origin). 
The  above  formula  then  gives  the  projection  with  its  proper 
sign. 

Thus,  the  segment  OP  (Fig.  106)  from  the  origin  to  any 
point  P(x,  y,  z)  can  be  regarded  as  a  vector  OP.  Its  projec- 
tions on  the  axes  of  coordinates  are 
the  coordinates  x,  y,  z  of  P.  These 
projections  are  also  called  the  rec- 
tangular components  of  the  vector  OP, 
and  OP  is  called  the  resultant  of  the 
components  OQ',  OQ',  OQ'",  or  also 
of  OQ',  Q'P"',  P"'P, 

/**  Fir     108 

Similarly,  in  Fig.  108,  if  P^P^  be 
regarded  as  a  vector,  the  projections  of  this  vector  P^P-i  on 
the  axes  of  coordinates  are  the  coordinate  differences  x^  —  Xy, 
2/2  —  2/u  22  —  ^i-     See  §  203. 

199.  Resultant.  The  proposition  of  §  19  that  the  sum  of 
the  projections  of  the  sides  of  an  open  polygon  on  any  axis  is 


X,§200] 


COORDINATES 


193 


equal  to  the  projection  of  the  closing  side  on  the  same  axis  and 
that  of  §  20  that  the  projection  of  the  resultant  is  equal  to  the 
sum  of  the  projections  of  its  components  are  readily  seen  to  hold 
in  three  dimensions  as  well  as  in  the  plane.  Analytically 
these  propositions  follow  by  considering  that  whatever  the 
points  Pi(.Ti,  y,,  z,),  P^(x^,  y^,  z^),  •••  P„(a;„ ,  2/„,  z^)  in  space, 
the  sum  of  the  projections  of  the  vectors  P1P2 ,  PjA  >  ••*  Pn-i^n 
on  the  axis  Ox  is  : 


(x2-x^)-{-(xs-X2)-{ \-(x^ 


00 „ 1  )  —  Vu„  "^  Out 


where  the  right-hand  member  is  the  projection  of  the  closing 
side  or  resultant  PiP„  on  Ox.  Any  line  can  of  course  be  taken 
as  axis  Ox. 

200.  Division  Ratio.  Two  points  Pi(xi,  y^,  z^  and 
Pi  (^2  J  2/2  ?  ^2)  beirig  given  by  their 
coordinates,  the  coordinates  x,  y,  z 
of  any  point  P  of  the  line  P^P^ 
can  he  found  if  the  division  ratio 
PyP/P^P^  =  k  is  known  in  luhich 
the  point  P  divides  the  segment 
P,P,  (Fig  109). 

Let  Qi,  Q,  ^2^36  the  projections 

of  Pi,  P,  P2  on  the  axis   Ox\  as 

Q  divides  Q^Q^  in  the  same  ratio  k  in  which  P  divides  PiPj, 

we  have  as  in  §  3  : 

x=x^-\-  k{x2  —  x{). 

Similarly  we  find  by  projecting  on  Oy,  Oz : 

y  =  yi  +  ^Cv2 - Vi),  z  =  z,-{- k{z. - z^. 

If  k  is  positive,  P  lies  on  the  same  side  of  Pi  as  does  P2;  if 
k  is  negative,  P  lies  on  the  opposite  side  of  Px  (§  3). 
o 


Fig.  109 


194  SOLID  ANALYTIC  GEOMETRY  [X,  §  201 

201.  Direction  Cosines.  Instead  of  using  the  cartesian 
coordinates  a,  y^  z  to  locate  a  point  P  (Fig.  110)  we  can  also 
use  its  radius  vector  r  =  OP,  i.e.  the  length  of  the  vector  drawn 
from  the  origin  to  the  point,  and  its  direction  cosines,  i.e.  the 
cosines  of  the  angles  a,  p,  y,  made 
by  the  vector  OP  with  the  axes  Ox, 
Oy,  Oz.     We  have  evidently 

05  =  r  cos  a,  y  =  r  cos^,  z  =  r  cosy. 

As  a  line  has  two  opposite  senses 
we    can    take    as    direction    cosines    j 
of  any   line   parallel    to    OP    either 
cos  a,  cos  p,  cos  y,  or  —  cos  a,  —  cos  p,  —  cos  y. 

The  direction  cosines  cos  a,  cos  )8,  cos  y  of  a  vector  OP  are 
often  denoted  briefly  by  the  letters  I,  m,  n,  respectively,  so 
that  the  coordinates  of  P  are 

x  =  lr,  y  =  mr,  z  =  nr. 

The  direction  cosines  of  any  parallel  line  are  then  Z,  m,  n 
or  —  ?,  —  m,  —  n. 

202.  Pythagorean  Relation.  Tlie  sum  of  the  squares  of  the 
direction  cosines  of  any  line  is  equal  to  one. 

For  the  equations  of  §  201  give  upon  squaring  and  adding, 

since  a^ -\- y'^ -\- z^  =  r^ : 

cos'^a  +  cos*  P  +  cos*  7  =  1* 
or 

P  +  m2  +  n2  =  1 ; 

and  this  still  holds  when  I,  m,  n  are  replaced  by  —l,—m,—  n. 
Since  this  result  is  derived  directly  from  the  Pythagorean 
Theorem  of  geometry,  it  may  be  called  the  Pythagorean  Pela- 
tion  between  the  direction  cosines.  Notice  that  I,  m,  n  can  be 
regarded  as  the  coordinates  of  the  extremity  of  a  vector  of 
unit  length  drawn  from  the  origin  parallel  to  the  line. 


X,  §  202]  COORDINATES  195 

EXERCISES 

1.  Find  the  length  of  the  radius  vector  and  its  direction  cosines  for 
each  of  the  following  points  :  (5,  —  3,  2);  (—  3,  —  2,  1);  (—4,  0,  8). 

2.  The  direction  cosines  of  a  line  are  proportional  to  1,  2,  3 ;  find 
their  values. 

3.  A  straight  line  makes  an  angle  of  30°  with  the  axis  Ox  and  an 
angle  of  60°  with  the  axis  Oy  ;  what  is  the  third  direction  angle  ? 

4.  What  is  the  direction  of  a  line  when  Z  =  0  ?  when  Z  =  m  =  0  ? 

5.  What  are  the  direction  cosines  of  that  line  whose  direction  angles 
are  equal  ? 

6.  What  are  the  direction  cosines  of  the  line  bisecting  the  angle 
between  two  intersecting  lines  whose  direction  cosines  are  Z,  m,  n  and  Z', 
wi',  n'^  respectively  ? 

7.  Find  the  direction  cosines  of  the  line  which  bisects  the  angle 
between  the  radii  vectoresof  the  points  (3,  —  4,  2)  and  (—  1,  2,  3). 

8.  Three  vertices  of  a  parallelogram  are  (4,  3,  —2),  (7,  —1,  4), 
(—  2,  1,  —  4);  find  the  coordinates  of  the  fourth  vertex  (three  solutions). 

9.  In  what  ratio  is  the  line  drawn  from  the  point  (2,  —  5,  8)  to  the 
point  (4,  6,  —  2)  divided  by  the  plane  Ozx  ?  by  the  plane  Oxy  ?  At  what 
points  does  this  line  pierce  these  coordinate  planes  ? 

10.  In  what  ratio  is  the  line  drawn  from  the  point  (0,  5,  0)  to  the 
point  (8,  0,  0)  divided  by  the  line  in  the  plane  Oxy  which  bisects  the 
angle  between  the  axes  ? 

11.  Find  the  coordinates  of  the  midpoint  of  the  line  joining  the  points 
(4,  —3,  8)  and  (6,  5,  —  9).    Find  the  points  which  trisect  the  same  segment. 

12.  If  we  add  to  the  segment  joining  the  points  (4,  1,  2)  and  (—  2, 
5,  7)  a  segment  of  twice  its  length  in  each  direction,  what  are  the  coordi- 
nates of  the  end  points  ? 

13.  Find  the  coordinates  of  the  intersection  of  the  medians  of  the  tri- 
angle whose  vertices  are  Pi(a;i,  yi ,  ^i),  P2{X2 ,  2/2,  Zi)  ^  Pz{xz ,  2/3 ,  ^z)- 

14.  Show  that  the  lines  joining  the  midpoints  of  the  opposite  edges  of 
a  tetrahedron  intersect  and  are  bisected  by  their  common  point. 

15.  Show  that  the  projection  of  the  radius  vector  of  the  point 
P(a;,  ?/,  z)  on  a  line  whose  direction  cosines  are  V,  m',  n'  is  Vx-\-m'y-\-n'z. 


196 


SOLID  ANALYTIC  GEOMETRY 


[X,  §  203 


M 


(T 


Fig.  Ill 


203.  Projections.  Components  of  a  Vector.  If  two  points 
-f\(^ij  2/i  >  ^\)  and  Piix^^  y^,  z^  are  given  by  their  coordinates, 
the  projections  of  the  vector,  P^P^  on 
the  axes,  or  what  amounts  to  the 
same,  on  parallels  to  the  axes  drawn 
through  Pi  (Fig.  Ill),  are  evidently 
(§  198)  : 

P^q  =  x^-x^,  PiR^Vi  —  yi, 
P^S  =  z^-z,. 

These  projections,  or  also  the  vectors 

P\Q,  QNfNPz,  are  called  the  rectangular  components  of  the 

vector  P1P2 ,  or  its  components  along  the  axes. 

If  d  is  the  length  of  the  segment  P^P^ ,  its  direction  cosines  Z, 
m,  n  are,  since  P2Q  is  perpendicular  to  PiQ,  P^R  to  P^R,  P^S 
to  P,S: 

J  Xn  —  X,  ?/,   —   Vl  2;.,  —  Z, 

a  a  a 

These  relations  can  also  be  written  in  the  form : 

^2  -  a?i  ^  .V2  -  Vi^ Zj-Zi^  ^ 
I  m  n 


flt,mt.iit) 


204.  Angle  between  Two  Lines.  If  the  directions  of  two  lines 
are  given  by  their  direction  cosines  l^  ?Wi,  ?^l  ajid  h,  Wo,  tio,  the 
angle  ij/  between  the  two  lines  is  given 
by  the  formula 

cos  <|/  =  lih  +  inxnii  +  nin2 . 

For,  drawing  through  the  origin 
two  lines  of  direction  cosines  ^ ,  wij , 
Til  ^-rid  I2,  m^,  W2  3-nd  taking  on  the 
former  a  vector  OPi  of  unit  length,  Fia.  112 

the  projection  OP  of  OPi  on  the  other  line  is  equal  to  the 


X,  §  206]  COORDINATES  197 

cosine  of  the  required  angle  if/.  On  the  other  hand,  OPi  has 
li,  rrii ,  rii  as  components  along  the  axes  ;  hence,  by  §  199 : 

cos  k{/  =  Zi?2  4-  wi^mg  +  riiWa  • 
Two  intersecting  lines  (or  any  two  parallels  to  them)  make 
two  angles,  say  ij/  and  -rr—ip.  But  if  the  direction  cosines  of 
each  line  are  given,  a  definite  sense  has  been  assigned  to  each 
line,  and  the  angle  between  the  lines  is  understood  to  be  the 
angle  between  these  senses. 

205.  Conditions  for  Parallelism  and  for  Perpendicularity. 

If,  in  particular,  the  lines  are  parallel,  we  have  either  li^h, 
mi  =  m2 ,  ?ii  =  ^2 ,  or  Z^  =  —  ?,,  m^  =  —  m2 ,  w i  =  —  Wg ;  hence  in 
either  case  ^i  _  '^i  _  ^ 

I2     m2     ^2 
This  then  is  the  condition  of  parallelism  of  two  lines  whose 
direction  cosines  are  li,  rrii,  %  and  I2,  mg,  %2- 

If  the  lines  are  perpendicular,  i.e.  ii  i{/  =  -|-  tt,  we  have 
cos  j/^  =  0  ;  hence  the  condition  of  perpendicularity  of  two  lines 
whose  direction  cosines  are  l^,  m^,  n^  and  I2 ,  m2 ,  ^2  is 

hh  +  Wim2  +  nin2  =  0. 

206.  The  formula  of  §  204  gives 

sin2  ^  =  1  —  cos2 1//  =  1  —(hl2  +  Wim2  +  nin2)^. 
As  (§  202)  (Zi2  +  mi2  +  ni^)(ih^  +  m-P-  +  n-?^^  1,  we  can  write  this  ex- 
pression in  the  form 

Zi2  +  m^  +  n^  hh  +  mirm  +  fiiUi 

hh  +  mim2  +  win2    h^  +  m2^+  n-^ 


sin^i// 


which  can  also  be  expressed  as  follows 


sin^  i//  = 


mo    712 


h    wi 


711     h 
«2     h 

The  direction  (I,  m,  n)  perpendicular  to  two  given  different  directions 
(li ,  mi ,  ni)  and  (^2,  «i2,  n^)  is  found  by  solving  the  equations  (§  205) 
hi  +  mim  +  Win  =  0, 
ZgZ  +  7W2WI  +  n2n  =  0, 


198 


SOLID  ANALYTIC  GEOMETRY         [X,  §  206 


I 

m 

n 

Hi      h 

whence 


If  we  denote  by  k  the  common  value  of  these  ratios,  we  have 


1  = 


wii    ni 


k, 


m 


Til     h 
112    h 


w=  \k; 

12      Wl2 I 


substituting  these  values  in  the  relation  (§  202)  l^  -\- m^  -{■  n^  =  1,  and 
observing  the  preceding  value  of  sin^,  we  find: 


TOi    ni 

m2      7121 


n2    li 


n=± 


h    wii 
I2    wtg 


sin^ 


sin  ^  sin  yj/ 

where  V  is  the  angle  between  the  given  directions. 

207.   Three  directions  (h ,  w»i ,  ni),  (Z2,  WI2,  W2),  (^3 ,  ^3 ,  ws)  are  com- 
planar,  i.e.  parallel  to  the  same  plane,  if  there  exists  a  direction  (/,  m,  n) 
perpendicular  to  all  three.    This  will  be  the  case  if  the  equations 
hi  +  fnim  +  wiw  =  0, 
hi  +  wi2m  4-  712/1  =  0, 
hi  +  w»3W»  +  nzn  =  0 
have  solutions  not  all  zero  ;  hence  the  condition  of  complanarity 
h    wii     Wi 
h    WI2    W2 
I3    WI3    ns 


=  0. 


EXERCISES 

1.  Find  the  length  and  direction  cosines  of  the  vector  drawn  from  the 
point  (5,  —2,  1)  to  the  point  (4,  8,  —  6)  ;  from  the  point  (a,  b,  c)  to  the 
point  (—a,  —b,  —c)  ;  from  (  —  a,  —6,  — c)  to  (a,  6,  c). 

2.  Show  that  when  two  lines  with  direction  cosines  Z,  w,  n  and 
Z',  w*',  n',  respectively,  are  parallel,  IV  +  mm'  +  nn'  =  ±  1. 

3.  Show  that  when  two  lines  with  direction  cosines  proportional  to 
a,  6,  c  and  a',  6',  c'  are  perpendicular  aa'  +  66'+  cc'  =  0  ;  and  when  the 
lines  are  parallel  a/a'  =  6/6'  =  c/c'. 

4.  Show  that  the  points  (5,  2,  -3),  (6,  1,  4),  (-2,  -3,  6), 
(—1,  —  4,  13)  are  the  vertices  of  a  parallelogram. 


X,  §  208]  COORDINATES  199 

6.  Show  by  direction  cosines  that  the  pomts  (6,  -3,  5),  (8,  2,  2), 
(4,  —8,  8)  lie  in  a  line. 

6.  Find  the  angle  between  the  vectors  from  (5,  8,  —  2)  to  (—2,  6,-1) 
and  from  (8,  3,  5)  to  (1,  1,  -6). 

7.  Find  the  angles  of  the  triangle  whose  vertices  are  (5,  2,  1), 
(0,3,  -1),(2,  -1,7). 

8.  Find  the  direction  cosines  of  a  line  which  is  perpendicular  to  two 
lines  whose  direction  cosines  are  proportional  to  2,  —3,  4,  and  6,  2,  —1, 
respectively. 

9.  Derive  the  formula  of  §  204  by  taking  on  each  line  a  vector  of  unit 
length,  OPi  and  OP2,  and  expressing  the  distance  P1P2  first  by  the 
cosine  law  of  trigonometry,  then  by  §  196,  and  equating  these  expressions. 

10.  Find  the  rectangular  components  of  a  force  of  12  lb.  acting  along 

a  line  inclined  at  60°  to  Ox  and  at  45°  to  Oy. 

11.  Find  the  resultant  of  the  forces  OPi ,  OP2 ,  OP3 ,  OP4  if  the  co- 
ordinates of  Pi,  P2,  P3,  P4,  with  Oas  origin,  are  (3,  —1,  2),  (2,  2,-1), 
(-1,2,  1),  (-2,  3,  -4). 

12.  If  any  number  of  vectors,  applied  at  the  origin,  are  given  by  the 
coordinates  x,  y,  z  of  their  extremities,  the  length  of  the  resultant  B  is 
V'(Sx)2  +  (Sy)'-^  +  (S2:)2   (see  Ex.  9,  p.  20),  and  its  direction  cosines 

are  2x/i?,  Sy/iJ,  ^zjB. 

13.  A  particle  at  one  vertex  of  a  cube  is  acted  upon  by  seven  forces 
represented  by  the  vectors  from  the  particle  to  the  other  seven  vertices ; 
find  the  magnitude  (length)  and  direction  of  the  resultant. 

14.  If  four  forces  acting  on  a  particle  are  parallel  and  proportional  to 
the  sides  of  a  quadrilateral,  the  forces  are  in  equilibrium,  i.e.  their  resultant 
is  zero.     Similarly  for  any  closed  polygon. 

208.  Translation  of  Coordinate  Trihedral.  Let  x,  y,  z  be 
the  coordinates  of  any  point  P  with  respect  to  the  trihedral 
formed  by  the  axes  Ox,  Oy,  Oz  (Fig.  113).  If  parallel  axes 
Oi^i,  Oi?yi,  O^Zi  be  drawn  through  any  point  Oi(a,  b,  c),  and  if 
^ij  Vi,  ^1  are  the  coordinates  of  P  with  respect  to  the  new  tri- 


200 


SOLID  ANALYTIC  GEOMETRY 


[X,  §  208 


hedral  OiXiy^Zi,  then  the  relations  between  the  old  coordinates 
X,  y,  z,  and  the  new  coordinates  «i ,  2/1,  ^i  of  one  and  the  same 
point  P  are  evidently 

x  =  a-{-x^,      y  =  b-\-yi,     z  =  c-{-Zi. 

The  coordinate  trihedral  has  thus 
been  given  a  translation,  represented 
by  the  vector  00^.     This  operation 


I 


^1  ic        ! 


is    also   called   a   transformation   to        X^ — •«- > 


I  I 


parallel  axes  through  Oj.  Fio.  113 

209.  Area  of  a  Triangle.  Any  two  vectors  OPi ,  OP2  drawn  from 
the  origin  determine  a  triangle  OPxPi,  whose  area  A  can  easily  be  ex- 
pressed if  the  lengths  ri ,  r2  and  direction  cosines 
of  the  vectors  are  given.  For,  denoting  the  angle 
P1OP2  by  ^,  we  have  for  the  area  A  : 

A  =  \  ViTi  sin  \{/^ 

where  sin  ^  can  be  expressed  in  terms  of  the  direc- 
tion cosines  by  §  206. 


% 

FiQ.  114 
210.    Moment  of  a  Force.    Such  areas  are  used  in  mechanics  to 

represent  the  moments  of  forces.    The  moment  of  a  force  about  a  point  O 

is  defined  as  the  product  of  the  force  into  the 

perpendicular  distance  of  0  from  the  line  of 

action  of  the  force.     Thus,  if  the  vector  P1P2 

(Fig.   115)  represent  a  force   (in  magnitude, 

direction,  and  sense)  the  moment  of  this  force   ^L.,,,,^^^^ 

about  the  origin  O  is  equal  to  twice  the  area 

of  the  triangle  OP1P2,  i.e.  to  the  area  of  the 

parallelogram  OP1P2P3 ,  where  OP3  is  a  vector 

equal  to  the  vector  P1P2.  Fig.  116 

It  is  often  more  convenient  to  represent  this  moment  not  by  such  an 

area,  but  by  a  vector  OQ^  drawn  from  O  at  right  angles  to  the  triangle, 

and  of  a  length  equal  to  the  number  that  represents  the  moment.     If  the 

body  on  which  the  force  acts  could  turn  freely  about  this  perpendicular, 

the  moment  would  represent  the  turning  effect  of  the  force  P1P2. 


X,  §  211] 


COORDINATES 


201 


The  sense  of  this  vector  that  represents  the  moment  is  taken  so  as  to 
make  the  vector  point  toward  that  side  of  the  plane  of  the  triangle  from 
which  the  force  P1P2  is  seen  to  turn  counterclockwise. 

211.  If  we  square  the  expression  found  in  §  209  for  the  area  of  the 
triangle  OP1P2  and  substitute  for  sin^  xp  its  value  from  §  206,  we  find  : 


^2 


=  iriW( 


mi     Hi 


+ 


712      h 


+ 


Zi     Wi 


1 


Hence  A^  is  the  sum  of  the  squares  of  the  three  quantities 


Ax  =  \  ri?'2 


mi    Wi 
mi    712 


,    Ay  =  I  rir2 


7ii    h 

7l2      h 


Inri 


h    «i2 


which  have  a  simple  geometrical  and  mechanical  interpretation.     For,  as 
the  coordinates  of  Pi ,  P2  are 


we  have,  e.g., 


Xi  =  hri,      yx  =  wiri,     zi  =  win, 

X2  =  hr2,        y2  =  W2r2,      Z2  =  n2^2» 


A,  =  l 


hn    min 

?2^2      W2»*2 

=  i 

xi    yi 

X2      2/2 

and  as  Xi ,  2/1  and  X2 ,  2/2  are  the  coordinates  of  the  projections  Qi ,  Q2  of 
Pi ,  P2  on  the  plane  Oxy,  Ag  represents  (§  12)  the  area  of  the  triangle 
0QiQ2,  ie.  the  projectio7i  on  the  plane  Oxy  of  the  area  OP1P2.  Sim- 
ilarly, Ax  and  Ay  are  the  projections  of  the  area  OP1P2  on  the  planes 
Oyz  and  Ozx,  respectively.  As  any  three  mutually  rectangular  planes 
can  be  taken  as  coordinate  trihedrals,  our  formula  A^  =  A^  -\-  A^  +  A^ 
means  that  the  square  of  the  area  of  any  triangle  is  equal  to  the  sum  of 
the  squares  of  its  projections  on  any  three  mutually  rectangular  planes. 

In  mechanics,  2Ag  is  the  moment  of  the  projection  ^1^2  of  the  force 
P1P2  about  0,  or  what  is  by  definition  the  same  thing,  the  moment  of 
P1P2  about  the  axis  Oz.  Similarly,  for  2A^,  2  Ay .  The  proposition 
means,  therefore,  that  the  moments  of  P1P2  about  the  axes  Ox,  Oy,  Oz 
laid  off  as  vectors  along  these  axes  can  be  regarded  as  the  rectangular 
components  of  the  moment  of  Pi  P2  about  the  point  0  ;  in  other  words, 
2Agy  2  Ay,  2  A,  are  the  components  along  Ox,  Oy,  Oz  of  that  vector 
2  ^  (§  210)  which  represents  the  moment  of  P1P2  about  0. 


202 


SOLID  ANALYTIC  GEOMETRY  [X,  §  212 


Fio.  116 


212.  Polar  Coordinates.  The  position  of  any  point  P  (Fig. 
116)  can  also  be  assigned  by  its 
radius  vector  0P=  r,  i.e.  the  dis- 
tance of  P  from  a  fixed  origin  or 
pole  0,  and  two  angles :  the  colati- 
tude  $,  i.e.  the  angle  NOP  made 
by  OP  with  a  fixed  axis  ON,  the 
X>olar  axis,  and  the  longitude  <f>, 
i.e.  the  angle  AOP  made  by  the 
plane  of  $  with  a  fixed  plane 
NOA  through  the  polar  axis,  the 
initial  meridian  plane. 

A  given  radius  vector  r  confines  the  point  P  to  the  sphere 
of  radius  r  about  the  pole  0.  The  angles  6  and  <^  serve  to 
determine  the  position  of  P  on  this  sphere.  This  is  done  as 
on  the  earth's  surface  except  that  instead  of  the  latitude,  which 
is  the  angle  made  by  the  radius  vector  with  the  plane  of  the 
equator  AP',  we  use  the  colatitude  or  polar  distance  6  =  NOP. 

The  quantities  r,  6,  and  <^  are  the  j^olar  or  spherical  coordi- 
nates of  P.  After  assuming  a  point  0  as  pole,  a  line  ON 
through  0,  with  a  definite  sense,  as  polar  axis,  and  a  (half-) 
plane  through  this  axis  as  initial  meridian  plane,  every  point 
P  has  a  definite  radius  vector  r  (varying  from  zero  to  infinity), 
colatitude  $  (varying  from  0  to  -n),  and  a  definite  longitude  <^ 
(varying  from  0  to  2  it).  The  counterclockwise  sense  of  rotation 
about  the  polar  axis  is  taken  as  the  positive  sense  of  <^. 


213.  Transformation  from  Cartesian  to  Polar  Coordinates. 

The  relations  between  the  cartesian  coordinates  x,  y,  z  and  the 
polar  coordinates  r,  6,  <f>  of  any  point  P  appear  directly  from 
Fig.  117.  If  the  axis  Oz  coincides  with  the  polar  axis,  the 
plane  Oxy  with  the  equatorial  plane,  i.e.  the  plane  through  the 


X,  §  2131  COORDINATES  203 

pole  at  right  angles  to  the  polar  axis,  while  the  plane  Ozx  is 
taken  as  initial  meridian  plane,  the  pro-  ' 

jections  of  OP=r  on  the  axis  Oz  and  * 

on  the  equatorial  plane  are 

OIi  =  rGose,  OQ  =  r sine. 


Projecting  OQ  on  the  axes  Ox,  Oy,we   -/^ y H& 

find  Fig,  117 

a;  =  r  sin  ^  cos  <^,     ?/  =  r  sin  ^  sin  </>,     z  =  rcos0. 

Also     r  =  Va^  +  2/^  +  2-,  cos  (9  = ^  tan<^  =  ^. 

Vaj2  +  y'^-\-z^  » 

EXERCISES 

1.  Find  the  area  of  the  triangle  whose  vertices  are  (a,  0,  0),  (0,  6,  0), 
(0,  0,  c). 

2.  Find  the  area  of  the  triangle  whose  vertices  are  the  origin  and  the 
points  (3,  4,  7),  (-  1,  2,  4). 

3.  Find  the  area  of  the  triangle  whose  vertices  are  (4,  —  3,  2), 
(6,4,4),  (-5,  -2,  8). 

4.  The  cartesian  coordinates  of  a  point  are  1,  V3,  2  V3  ;  what  are  its 
polar  coordinates  ? 

5.  If  r  =  5,  fl  =  i  TT,  0  =  I TT,  what  are  the  cartesian  coordinates  ? 

6.  The  earth  being  taken  as  a  sphere  of  radius  3962  miles,  what  are 
the  polar  and  cartesian  coordinates  of  a  point  on  the  surface  in  lat.  42°  17' 
N.  and  long.  83°  44'  W.  of  Greenwich,  the  north  polar  axis  being  the  axis 
Oz  and  the  initial  meridian  passing  through  Greenwich  ?  What  is  the 
distance  of  this  point  from  the  earth's  axis  ? 

7.  Find  the  area  of  the  triangle  whose  vertices  are  (0, 0,  0) ,  (ri ,  ^1 ,  0i) , 
(r2,  ^2,  02). 

8.  Express  the  distance  between  any  two  points  in  polar  coordinates. 

9.  Find  the  area  of  any  triangle  when  the  cartesian  coordinates  of  the 
vertices  are  given. 

10.  Find  the  rectangular  components  of  the  moment  about  the  origin 
of  the  vector  drawn  from  (1,  —  2,  3)  to  (3,  1,  —  1). 


CHAPTER   XI 

THE  PLANE   AND   THE   STRAIGHT  LINE 

PART   I.     THE   PLANE 

214.  Locus  of  One  Equation.  In  plane  analytic  geometry 
any  equation  between  the  coordinates  x,  y  ov  r,  <f>  of  a.  point  in 
general  represents  a  plane  curve.  In  particular,  an  equation  of 
the  first  degree  in  x  and  y  represents  a  straight  line  (§  30) ; 
an  equation  of  the  second  degree  in  x  and  y  in  general  repre- 
sents a  conic  section  (§154). 

In  solid  analytic  geometry  any  equation  between  the  coordi- 
nates X,  y,  z  or  r,  0,  4>  of  a.  point  in  general  represents  a  surface. 
Thus,  if  any  equation  in  Xj  y,  z, 

F{x,y,z)^0, 

be  imagined  solved  for  z  so  as  to  take  the  form 

2!=/(a;,y), 

we  can  find  from  this  equation  to  every  point  (a?,  y)  in  the 
plane  Oxy  one  or  more  ordinates  z  (which  may  of  course  be 
real  or  imaginary),  and  the  locus  formed  by  the  extremities  of 
the  real  ordinates  will  in  general  form  a  surface.  It  may  how- 
ever happen  in  particidar  cases  that  the  locus  of  the  equation 
F(Xy  y,  z)  =  0,  i.e.  the  totality  of  all  those  points  whose  coordi- 
nates X,  y,  z  when  substituted  in  the  equation  satisfy  it,  con- 
sists only  of  isolated  points,  or  forms  a  curve,  or  that  there  are 
no  real  points  satisfying  the  equation. 

Similar    considerations    apply    to    an    equation    in    polar 

coordinates 

F(r,  $,<(>)  =0, 

204 


XI,  §  216]  THE  PLANE  205 

215.  Locus  of  Two  Simultaneous  Equations.  Two  simulta- 
neous equations  in  cc,  y,  z  (or  in  the  polar  coordinates  r,  6,  <f>) 
will  in  general  represent  a  curve  in  space,  namely,  the  inter- 
section of  the  two  surfaces  represented  by  the  two  equations 
separately. 

Thus,  in  the  present  chapter,  we  shall  see  that  an  equation  of 
the  first  degree  in  x,  y,  z  represents  a  plane  and  that  therefore 
two  such  equations  represent  a  straight  line,  the  intersection  or 
the  two  planes.  In  chapters  XII  and  XIII  we  shall  discuss 
loci  represented  by  equations  of  the  second  degree,  which  are 
called  quadric  surfaces. 

216.  Equation  of  a  Plane.  Every  equation  of  the  first  degree 
in  X,  y,  z  represents  a  plane.  The  plane  is  defined  as  a  surface 
such  that  the  line  joining  any  two  of  its  points  lies  completely 
in  the  surface.  We  have  therefore  to  show  that  if  the  general 
equation  of  the  first  degree 

(1)  Ax-{-By-^Cz  +  D=0 

is  satisfied  by  the  coordinates  of  any  two  points  Pi(aJi,  ?/i,  Zj) 

and  P^ix.^ ,  2/2  J  %)?  i-^-  if 

Ax^  -f  Byi  +  (7%  +  J5  =  0, 


^  -^  ^  Ax2  +  By2-\-Cz2  +  D=:0, 

then    (1)    is    satisfied    by   the    coordinates    of    every    point 
P(x,  y,  z)  of  the  line  PiP2- 

Now,  by  §  200,  the  coordinates  of  every  point  of  the  line 
P1P2  can  be  expressed  in  the  form 

x  =  Xi-\- k(x2  —  a^i),  y  =  yi-\- Kv^  —  Vi),  z  =  ^i  +  K^2  —  %)» 
where  k  is  the  ratio  in  which  P  divides  PiP^,  i.e. 

h  =  P,P/P^P,. 
We  have  therefore  to  show  that 
A[x^  +  K^2  -  a^)]  +  ^[2/1  +^fe-2/i)]  +  C[_z^+K^2-Zi)']  +^=0, 


206  SOLID  ANALYTIC  GEOMETRY        [XI,  §  216 

■whatever  the  value  of  k.     Adding  and  subtracting  kD,  we  can 
write  this  equation  in  the  form 
{l-k){Ax^  +  By^+Cz^-\-D)-^k{Ax^-\-By^+Cz^  +  U)=zO', 

and  this  is  evidently  true  for  any  k,  owing  to  the  conditions  (2). 

217.  Essential  Constants.  The  equation  (1)  will  still  rep- 
resent the  same  plane  when  multiplied  by  any  constant  differ- 
ent from  zero.  Since  A,  B,  C  cannot  all  three  be  zero,  we 
can  divide  (1)  by  one  of  these  constants ;  it  will  then  contain 
not  more  than  three  arbitrary  constants.  We  say  therefore 
that  the  general  equation  of  a  plane  contains  tJwee  essential 
constants.  This  corresponds  to  the  geometrical  fact  that  a 
plane  can,  in  a  variety  of  ways,  be  determined  by  three  condi- 
tions, such  as  the  conditions  of  passing  through  three  points. 

218.  Special  Cases.  If,  in  equation  (1),  Z>  =  0,  the  plane 
evidently  passes  through  the  origin. 

If,  in  equation  (1),  C=0,  so  that  the  equation  is  of  the 
form  Ax  -[-  By  -\-  D  =  0,  this  equation  represents  the  plane 
perpendicular  to  the  plane  Oxy  and  passing  through  the  line 
whose  equation  in  the  plane  Oxy  is  Ax  -\-  By  -\-  D  =  0.  For, 
the  equation  Ax  +  J5?/  -|-  2)  =  0  is  satisfied  by  the  coordinates 
of  all  points  («,  ?/,  z)  whose  x  and  y  are  connected  by  the  re- 
lation Ax -\- By -{- D  =  0  and  whose  z  is  arbitrary,  but  it  is  not 
satisfied  by  the  coordinates  of  any  other  points.  Similarly,  if 
5  =  0  in  (1),  the  plane  is  perpendicular  to  Ozx;  if  ^  =  0,  the 
plane  is  perpendicular  to  Oyz. 

It  B  =  0  and  C  =  0  in  (1),  the  equation  obviously  represents 
a  plane  perpendicular  to  the  axis  Ox ;  and  similarly  when  0 
and  A,  or  A  and  B  are  zero. 

Notice  that  the  line  of  intersection  of  (1)  with  the  plane 
Oxy,  for  instance,  is  represented  by  the  simultaneous  equations 
Ax  +  By+Cz  +  D=0,   2  =  0. 


XI,  §  220] 


THE  PLANE 


207 


219.   Intercept  Form.     If  D  =^  0,  the  equation  (1)  can  be 
divided  by  Z) ;  it  then  assumes  the  form 

If  Aj  Bf  G  are  all  different  from  zero,  this  equation  can  be 
written 


+ 


1, 


-D/A  '  -D/B  '   -D/G 
or,  putting  -  D/A  =  a,  -  D/B  =  b,  -  D/G  =  c : 

(3) 


a     0     c 


In  this  equation,  called  the  intercept  form  of  the  equation 
of  a  plane,  the  constants  a,  h,  c  are  the  intercepts  made  by  the 
plane  on  the  axes  Ox,  Oy,  Oz  respectively.  For,  putting,  for 
instance,  2/  =  0  and  2:  =  0,  we  find  x  =  a\  etc. 

220.  Plane  through  Three  Points.     If  the  plane 

Ax  +  By  +  Gz-\-D  =  0 

is  to  pass  through  the  three  points  Pi(xi,  y^,  z^,  ^2(^2)  2^2 ?  ^2)) 
A  (^3  J  2/3?  ^z)j  the  three  conditions 

Ax,  +  By,  +  Gz,-\-D  =  0, 
Ax^-^By^-\-Gz^+D  =  0, 
Ax^  +  By^  +  Gz^  +  D  =  (i 

must  be  satisfied.  Eliminating  A,  B,  G,  D  between  the /owr 
preceding  equations,  as  in  §  55,  we  find  the  equation  of  the 
plane  passing  through  the  three  points  in  the  form 


X 

y 

z 

1 

X, 

2/1 

^l 

1 

x^ 

2/2 

2^2 

1 

X3 

2/3 

2=3 

1 

208 


SOLID  ANALYTIC  GEOMETRY       [XI,  §  220 


EXERCISES 

1.  Find  the  intercepts  made  by  the  following  planes  : 

(a)  4a;  + 12y +  3^  =  12;  (&)  15a:- 6y  +  10« -f  30  =  0  ; 

Qc)  x-y  ■}-z-l=0;  (rf)x+2y  +  30  +  4  =  O. 

2.  Interpret  the  following  equations : 

(a)a;  +  y  +  «  =  l;  {b)5y-3z  =  12, 

(c)  x  +  y=0;  (d)  5  y  +  12  =  0. 

3.  Find  the  plane  determined  by  the  points  (2,  1,  3),   (1,  —  5,  0), 
(4,6,  -1). 

4.  Write  down  the  equation  of  the  plane  whose  intercepts  are  3,  2,  —  5. 

6.   Find   the   intercepts  of    the    plane   passing    through   the   points 
(3,  -1,4),  (6,  2,  -3),  (-1,-2,  -3). 

6.  If  planes  are  parallel  to  and  a  distance  a  from  the  coordinate  planes, 
what  are  their  intercepts  ?    What  are  their  equations  ? 

7.  Show   that    the    four    points    (4,  3,  3),    (4,-3,-9),    (0,  0,  3), 
(2,  1,  2)  lie  in  a  plane  and  find  its  equation. 

221.   Normal   Form.     The  position  of  a  plane  in  space  is 
fully  determined  by  the  length  p  =  ON  (Fig.  118)  of  the  per- 
pendicular let  fall  from  the  origin 
on  the  plane  and  the  direction  co- 
sines I,  m,  n  of  this  perpendicular 
regarded  as  a  vector  ON.  Let  Pbe 
any  point  of  the  plane  and  OQ  =x, 
QR  =  y,  EP=  z  its  coordinates ;  as 
the  projection  of  the  open  polygon 
OQRP  on   ON   is   equal  to    ON 
(§  199)  we  have 
(4)  lx  +  my-\-  nz  =p. 

This  equation  is  called  the  normal  form  of  the  equation  of  a 
plane.  Observe  that  the  number  p  is  always  positive,  being 
the  distance  of  the  plane  from  the  origin,  or  the  length  of  the 
vector  ON.     Hence  Ix  +  my -{- nz  is  always  positive. 


FiQ.  118 


XI,  §222]  THE  PLANE  209 

222.  Reduction  to  the  Normal  Form.  The  equation  Ax  + 
By  +  Cz  -\-  D  =  0  is  in  general  not  of  the  form  ly-{-my-\-nz—p 
since  in  the  latter  equation  the  coefficients  of  x,  y,  z,  being  the 
direction  cosines  of  a  vector,  have  the  property  that  the  sum 
of  their  squares  is  equal  to  1,  while  A^  +  B^-\-  C^  is  in  general 
not  equal  to  1.  But  the  general  equation  can  be  reduced  to 
the  normal  form  by  multiplying  it  by  a  constant  factor  Ic 
properly  chosen.     The  equation 

TcAx  +  l<^By  +  hCz  -\-kD=0 
evidently  represents   the   same  plane  as   does   the   equation 
Ax  -i-  By  +  Cz  -\-  D  =  0;  and  we  can  select  Jc  so  that 

{JcAy  +  (JcBy  +  (kCy  =  l,     viz.  k.  ^ 


±VA^  +  B^+C^ 
As  in  the  normal  form  the  right-hand  member  p  is  positive 
(§  221)  the  sign  of  the  square  root  should  be  selected  so  that 
kD  becomes  negative. 

TTie  normal  form  is  therefore  obtained  by  dividing  the  equation 
Ax  +  By  -\-  Cz+D  =  Oby  ±  VA^  +  B^  +  C^  according  as  D  is 
negative  or  positive. 

It  follows  at  the  same  time  that  the  direction  cosines  of  any 
normal  to  the  plane  Ax  -\-  By  -{-  Cz  -{-  D  =  0  are  proportional 
to  A,  B,  C,  viz. 


I  z= ,    m  = 


C 


and  that  the  distance  of  the  plane  from  the  origin  is 

-D 

P  = 


±VAr-hB'-^0' 
the  upper  sign  of  the  square  root  to  be  used  when  D  is  nega- 
tive, the  lower  when  D  is  positive. 


210 


SOLID  ANALYTIC  GEOMETRY        [XI,  §  223 


223.   Distance  of  Point  from  Plane.     Let  Ix -\- my  +  nz  =  p 

be  the  equation  of  a  plane  in  the  normal  form,  Pi(iCi,  yi,  z^ 
any  point  not  on  this  plane  (Fig.  119).     The  projection  OS  of 
the  vector  OPi  on  the  normal  to  the 
plane  being  equal  to  the  sum  of  the 
projections  of  its  components  0Q  = 
Xi ,  QR  =  2/i,  RPi  =  2!i,  we  have 
OS  =  Za^  +  myx  +  nzj . 

Hence  the  distance  d  of  Pi  from  the 
plane,  which  is  equal  to  NS,  will  be 

d=  OS  —  0N=  Ixi  +  myi  +  7iZi  —  ;>.  fig.  ii9 

If  this  expression  is  negative,  the  point  Pj  lies  on  the  same 
side  of  the  plane  as  does  the  origin ;  if  it  is  positive,  the  point 
Pj  lies  on  the  opposite  side  of  the  plane.  Any  plane  thus  di- 
vides space  into  two  regions,  in  one  of  which  the  distance  of 
every  point  from  the  plane  is  positive,  while  in  the  other  the 
distance  is  negative.  If  the  plane  does  not  pass  through  the 
origin,  the  region  containing  the  origin  is  the  negative  region ; 
if  it  does,  either  side  can  be  taken  as  the  positive  side. 

To  find  the  distance  of  a  point  Pi(xi,  ?/i,  Zi)  from  a  plane 
given  in  the  general  form 

Ax-^By  +  Cz-hD  =  0, 
we  have   only  to  reduce  the   equation   to  the  normal   form 
(§  223)  and  then  to  substitute  for  x,  y,  z  the  coordinates  a^i,  3/1, 

Zi  of  Pi ;  thus 

^^Axi  +  Byi-{-  Czi  4-  D 

the  square  root  being  taken  with  +  or  —  according  as  Z)  is 
negative  or  positive. 

Notice  that  d  is  the  distance  from  the  plane  to  the  point 
Pi,  not  from  P^  to  the  plane. 


XI,  §225]  THE  PLANE  211 

224.  Angle  between  Two  Planes.  As  two  intersecting 
planes  make  two  angles  whose  sum  =  ir,  we  shall,  to  avoid  any 
ambiguity,  define  the  angle  between  the  planes  as  the  angle 
between  the  perpendiculars  (regarded  as  vectors)  drawn  from 
the  origin  to  the  two  planes. 

If  the  equations  of  the  planes  are  given  in  the  normal  form, 
liX-{-miy  +  niZ=pi, 
l^-{-m^  +  n.;Z  =P2i 
we  have,  by  §  204,  for  the  angle  i/^  between  the  planes : 

cos  \p  =  l-J.2  -f  mim2  +  n^a^ . 
If  the  equations  of  the  planes  are  in  the  general  form, 

we  find  by  reducing  to  the  normal  form  (§  222) : 


cosj/^  = 


A^A2-^B,B,+  C,C2 


±  V A'  +  B,'  +  0^2 .  ±  ^A,'  +  B,'  +  C,^ 

225.  Bisecting  Planes.  To  find  the  equations  of  the  two 
planes  that  bisect  the  angles  formed  by  two  intersecting  planes 
given  in  the  normal  form, 

liX-{-miy-{-niZ—pi  =  0,  l2X-{-m^ -{-n^z—pz^O, 

observe  that  for  any  point  in  either  bisecting  plane  its  distances 
from  the  two  given  planes  must  be  equal  in  absolute  value. 
Hence  the  equations  of  the  required  planes  are 

liX  +  rriiy  -f-  n^z  —pi=z±  (Icpc  -\-  m^y  -\-n^—  p^. 
To  distinguish  the  two  planes,  observe  that  for  the  plane  that 
bisects  that  pair  of  vertical  angles  which  contains  the  origin 
the  perpendicular  distances  are  in  the  one  angle  both  positive, 
in  the  other  both  negative;  hence  the  plus  sign  gives  this 
bisecting  plane. 


212  SOLID  ANALYTIC  GEOMETRY        [XI,  §  225 

If  the  equations  of  the  planes  are  given  in  the  general  form, 

first  reduce  the  equations  to  the  normal  form  (§  222). 

EXERCISES 

1.  A  line  is  drawn  from    the  origin  perpendicular   to  the   plane 
X  —  y  —  62  —  10  =  0;   what  are  the  direction  cosines  of  this  line ? 

2.  Find  the  distance  from  the  origin  to  the  plane  2x-\-2y  —  z=6. 

3.  Find  the  distances  of  the  following  planes  from  the  origin : 
(a)  3x-Ay-\-5z-S  =  0,  ^b)x  +  y+z  =  0, 
(c)  2y-50  =  3,                                         (d)  Sx-4y-\-6  =  0. 

4.  Find    the    distances   from    the   following    planes    to    the    point 
(2,1,  -3): 

(a)  3  X  +  6  y  -  6  2  =  8,       (6)  2x-Sy-z  =  0,       (c)  x  +  y -[- z  =  0. 
6.   Find  the  plane  through  the  point  (4,  8,  1)  which  is  perpendicular 
to  the  radius  vector  of  this  point ;  also  the  parallel  plane  whose  distance 
from  the  origin  is  10  and  in  the  same  sense. 

6.  Find  the  plane  through  the  point  (—1,2,  —  4)  that  is  parallel  to 
the  plane  4x  —  3y  +  22  =  8;  what  is  the  distance  between  these  planes  ? 

7.  Find  the  distance  between  the  planes  4x  —  6y  —  2z  =  6,  4x  —  5y 
-  2  «  +  8  =  0. 

8.  Are  the  points  (6,  1,  —  4)  and  (4,  —  2,  3)  on  the  same  side  of  the 
plane  2x +  Sy  -  6z -^  1  =0? 

9.  Write  down  the  equation  of  the  plane  equally  inclined  to  the  axes 
and  at  the  distance  p  from  the  origin. 

10.  Show  that  the  relation  between  the  distance  p  from  the  origin  to  a 
plane  and  the  intercepts  a,  6,  c  is  1/a^  +  l/b^  +  l/c^  =  1/p/^. 

11.  Show  that  the  locus  of  the  points  equally  distant  from  the  points 
-Pi(a;i ,  yi ,  2i)  and  P^ixo ,  j/a ,  22)  is  a  plane  that  bisects  P1P2  at  right 
angles. 

12.  Find  the  equations  of  the  planes  bisecting  the  angles  :  (a)  between 
the  planes  a:  +  2/  +  «-3=0,  2«-3y  +  42  +  3  =  0;  (6)  between  the 
planes  2x-2y-z  =  8,x  +  2y-2z  =  6. 


XI,  §  226] 


THE  PLANE 


213 


=  0. 


226.  Volume  of  a  Tetrahedron.  The  volume  of  the  tetrahe- 
dron whose  vertices  are  the  points  Pi(xi,  y^,  z{),  F^ixz,  y^,  z^, 
Pzi^zi  Vzi  ^z)i  A(^4>  2/4 >  ^a)  can  be  expressed  in  terms  of  the 
coordinates  of  the  points.  The  equation  of  the  plane  deter- 
mined by  the  points  P2,  P^,  P4  is  (§  220) 
X  y  z  1 
•^2  2/2  ^2  1 
•^3  2/3  %  1 
^•4    2/4     2:4     1 

Now  the  altitude  d  of  the  tetrahedron  is  the  distance  from  this 
plane  to  the  point  P^  (x^ ,  2/1 ,  ^i),  i-e.  (§  223) 

x^    2/1     2^1     1 
X2    2/2    ^2    -*- 

''^S       Vz       ^3        1 

X,      Va      z.      1 


2/2       ^2        1 

2 

Z2        X2        1 

2 

a;2    2/2    1 

2/3       2:3        1 

+ 

2^3       Xs       1 

+ 

Xz    2/3    1 

2/4       2^4       1 

24    a;4    1 

a?4    2^4     1 

d  = 


But  the  denominator  is  seen  immediately  to  represent  twice 
the  area  of  the  triangle  with  vertices  P2,  P3,  P4  (Ex.  9,  p.  203), 
i.e.  twice  the  base  of  the  tetrahedron.  Denoting  the  base  by  J5, 
we  then  have 


2Bd 


X, 

Vi 

2=1 

X2 

y2- 

2^2 

X, 

2/3 

2^3 

X, 

2/4 

2^4 

The  volume  of  the  tetrahedron  is  V=^Bd,  and  therefore 

Xi    2/1    2^1     1 
X2    2/2    2;,    1 

Xs      2/3      2:3      1 
2^4       2/4       2:4       1 


^ 


214  SOLID  ANALYTIC  GEOMETRY        [XI,  §  227 

227.   Simultaneous  Linear  Equations.    Two  simultaneous 
equations  of  the  first  degree, 

represent  in  general  the  line  of  intersection  of  the  two  planes 
represented  by  the  two  equations  separately.     For,  the  coordi- 
nates of  every  point  of  this  line,  and  those  of  no  other  point, 
satisfy  both  equations.     See  §  215  and  §§  231-232. 
Three  simultaneous  equations  of  the  first  degree, 

A,x  +  B,y+C,z  +  D,  =  0, 

A^x  +  ^2^  -h  Co2;  +  A  =  0, 

A,x  +  ^32/  +  C32;  -f  A  =  0, 
determine  in  general  the  point  of  intersection  of  the  three 
planes.  The  coordinates  of  this  point  are  found  by  solving 
the  three  equations  for  x,  y,  z.  But  it  may  happen  that  the 
three  planes  have  no  common  point,  as  when  the  three  lines  of 
intersection  are  parallel,  or  when  the  three  planes  are  parallel ; 
and  it  may  happen  that  the  planes  have  an  infinite  number  of 
points  in  common,  as  when  two  of  the  planes,  or  all  three, 
coincide,  or  when  the  three  planes  pass  through .  one  and  the 
same  line. 

Four  planes  will  in  general  have  no  point  in  common.     If  they  do,  i.e. 
if  there  exists  a  point  (xi  ,^1,^1)  satisfying  the  four  equations 

Aixi  +  BiVi  +  Cizi  +  Di  =  0, 

^23^1  +  Biyi  +  CiZi  +  Z>2  =  0, 

^33:1  +  ^3^1  +  CiZi  +  Da  =  0, 

AaXx  +  -84^1  +  CaZx  +  Z>4  =  0, 
we  can  eliminate  a;i,  j/i,  01,  1  between  these  equations  so  that  we  find 
the  condition 


=  0. 


Ax 

Bx 

Cx 

Dx 

A2 

B2 

C2 

i>2 

As 

Bz 

Cz 

Bz 

A4 

Bi 

c. 

B, 

XI,  §228]  THE  PLANE  215 

EXERCISES 

1.  Find  the  volume  of  the  tetrahedron  whose  vertices  are  (0,  0,  0), 
(o,  0,  0),  (0,  6,  0),  (0,  0,  c). 

2.  Find  the  volumes  of  the  tetrahedra  whose  vertices  are  the  following 
points : 

(a)  (7,  0,  6),  (3,  2,  1),  (-  1,  0,  4),  (3,  0,  -2). 

(6)  (3,  0,  1),   (0,  -  8,  2),   (4,  2,  0),  (0,  0,  10). 

(c)  (2,  1,-3),  (4,  -2,  1),  (3,  -7,  -4),  (5,  1,  8). 

3.  Find  the  coordinates  of  the  points  in  which  the  following  planes 
intersect : 

(a)  2x  +  6y  +  z-2  =  0,  xi-6y-{-z  =  0,  Sx-Sy -\-2  z  -  12  =  0, 
(6)  2x+y+z=a  +  b  +  G,  ^x-2  y-\-z=2  a-2b-\-c,  6x-y=3a-6. 

4.  Show  that  the  four  planes  6x  —  Sy—z  =  0,  4:X  —  2y  +  z  =  Sy 
Sx  +  2y  —  6z  =  6,  x  +  y  -\-  z  =  6  pass  through  the  same  point.  What 
are  the  coordinates  of  this  point  ? 

6.  Show  that  the  four  planes  4x  +  y  +  2!  +  4  =  0,  a;  +  2y  —  0  +  3  =  0, 
y  —  50  +  14=0,  x  +  y  -\-  z  —  2  =  0  have  a  common  point. 

6.  Show  that  the  locus  of  a  point  the  sura  of  whose  distances  from 
any  number  of  fixed  planes  is  constant  is  a  plane. 

228.  Pencil  of  Planes.  All  the  planes  that  pass  through 
one  and  the  same  line  are  said  to  form  a  pencil  of  planes,  and 
their  common  line  is  called  the  axis  of  the  pencil. 

If  the  equations  of  any  two  non-parallel  planes  are  given, 

say 

A,x  +  B,y  +  C,z  +  A  =  0, 

then  the  equation  of  any  other  plane  of  the  pencil  having  their 
intersection  as  axis  can  be  written  in  the  form 
(2)     {A,x  +  B,y  +  0,z  +  A)  +  K^^  +  ^22/  +  0,z  +  A)  =  0, 
where  A;  is  a  constant  whose  value  determines  the  position  of 
the  plane  in  the  pencil. 

For,  this  equation  (2)  being  of  the  first  degree  in  x,  y,  z 
certainly  represents  a  plane ;  and  the  coordinates  of  the  points 


216  SOLID  ANALYTIC  GEOMETRY        [XI,  §  228 

of  the  line  of  intersection  of  the  two  given  planes  (1),  since 
they  satisfy  each  of  the  equations  (1),  must  satisfy  the  equa- 
tion (2)  so  that  the  plane  (2)  passes  through  the  axis  of  the 
pencil. 

229.  Sheaf  of  Planes.  All  the  planes  that  pass  through 
one  and  the  same  point  are  said  to  form  a  sheaf  of  planes,  and 
their  common  point  is  called  the  center  of  the  sheaf. 

If  the  equations  of  any  three  planes,  not  of  the  same  pencil, 
are  given,  say 

A;,x  +  B^+C,z-^D^  =  0, 

then  the  equation  of  any  other  plane  of  the  sheaf  having  their 
point  of  intersection  as  center  can  be  written  in  the  form 
{A^x  +  B,y  +  C,z  4-  A)  +  K{Ax  +  Biy-irC^^-  A) 

+  k^{A^x-\-  B^  4-  C^z  +  A)  =  0, 

where   Ic^  and  k^  are  constants    whose  values   determine   the 
position  of  the  plane  in  the  sheaf. 
The  proof  is  similar  to  that  of  §  228. 

230.  Non-linear  Equations  Representing  Several  Planes. 

When  two  planes  are  given,  say 

Aa'  +  Ay  +  Ci2  +  A  =  0, 

A^  +  B^-\-C^  +  D^  =  0, 
then  the  equation 

{A^x  4-  B,y  -f  C,z  +  D,){A^x  +  5^2/  +  CjZ  -f  A)  =  0, 
obtained  by  equating  to  zero  the  product  of  the  left-hand  mem- 
bers (the  right-hand  members  being  reduced  to  zero),  is  satis- 
fied by  all  the  points  of  the  first  given  plane  as  well  as  all  the 
points  of  the  second  given  plane,  and  by  no  other  points. 

The  product  equation  is  therefore  said  to  represent  the  two 
given  planes.     The  equation  is  of  the  second  degree. 


XI,  §  230]  THE  PLANE  217 

Similarly,  by  equating  to  zero  the  product  of  the  left-hand 
members  of  the  equations  of  three  or  more  planes  (the  right- 
hand  members  being  zero)  we  obtain  a  single  equation  repre- 
senting all  these  planes.  An  equation  of  the  nth  degree  may, 
therefore,  represent  n  planes ;  it  will  do  so  if  its  left-hand  mem- 
ber can  be  resolved  into  n  linear  factors  with  real  coefficients. 

EXERCISES 

1.  Find  the  plane  that  passes  through  the  Ime  of  mtersection  of  the 
planes   5x  —  32/+4^  —  35=0,    x  +  y  —z  =  0   and  through  (4,  —  3,  2) . 

2.  Show  that  the  planes  3cc  —  2?/  +  5^  +  2=0,  x  +  y  —  0  —  5  =  0, 
6a;  +  ?/4-2;3—  13  =  0  belong  to  the  same  pencil: 

3.  Show  that  the  following  planes  belong  to  the  same  sheaf  and  find 
the  coordinates  of  the  center  of  the  sheaf :  6x  +  y  —  4;2  =  0,  x  +  2/  +  i2  =  5, 
2a;-4y-2  =  10,  2a;  +  3?/+;s  =  4. 

4.  What  planes  are  represented  by  the  following  equations  ? 

(a)  iK2-6x  +  8  =  0,  (6)  ?/2-9  =  0,  (c)   x'^-z^  =  0,  {d)  x'^-4xy  =  0. 

5.  Find  the  cosine  of  the  angle  between  the  following  pairs  of  planes : 
(a)  4:x-3y^z=6,  x-\-y-z=S  ;  (b)  2x+7  y-{-4z=2,  x-9y-2z=12. 

6.  Show  that  the  following  pairs  of  planes  are  either  parallel  or 
perpendicular : 

(a)  Sx-2y-\-5z=0,2x+Sy=8;  (b)  6x+2y-z=6,  lOx+iy-2  z=3; 
(c)  x  +  y-2z  =  S,  x-{-y+z=n;  (d)  x- 2y -  z  =  S,  3x -Qy-S z=6. 

7.  Find  the  plane  that  is  perpendicular  to  the  segment  joining  the 
points  (3,  —  4,  6)  and  (2,  1,  —  3)  at  its  midpoint. 

8.  Show  that  the  planes  Aix  +  Biy  +  Ciz  +  Di  =  0,  A2X  +  B2y  +  C2Z 
-f-i>2  =  0  are  parallel  (on  the  same  or  opposite  sides  of  the  origin)  if 

A1A2  +  B1B2  +  C1C2  ^^1^^ 

V^i2  -f-  Bi^  +  Ci2'  VA2'  +  B2^  +  CV 

9.  A  cube  whose  edges  have  the  length  a  is  referred  to  a  coordinate 
trihedral,  the  origin  being  taken  at  the  center  of  a  face  and  the  axes  par- 
allel to  the  edges  of  the  cube.     Find  the  equations  of  the  faces. 


218 


SOLID  ANALYTIC  GEOMETRY        [XI,  §  230 


a: 

y 

z 

1 

Xl 

y\ 

Z\ 

1 

X2 

yi 

22 

1 

A 

B 

c 

0 

10.  Show  that  the  plane  through  the  points  Pi  (xi ,  yi ,  Z\)  and 
P2fx2,  ?/2,  2^2)  and  perpendicular  to  the  plane  Ax -\-  By  ■\-  Cz -^^  D  —  ^ 
can  be  represented  by  the  equation 


=  0. 


11.  Find  those  planes  of  the  pencil  4a;  —  3y  +  52  =  8,  2x  +  3y  —  «  =  4 
which  are  perpendicular  to  the  coordinate  planes. 

12.  Find  the  plane  that  is  perpendicular  to  the  plane  2x  +  3y  —  «  =  1 
and  passes  through  the  points  (1,  1,  —  1),  (3,  4,  2). 

13.  Find  the  plane  that  is  perpendicular  to  the  planes  4x  —  3y  +  2  =  6, 
2a;  +  3y  —  60  =  4  and  passes  through  the  point  (4,  —  1,  6). 

14.  Show  that  the  conditions  that  three  planes  A\X + Bxy  -|-  C\Z + Di  =  0, 
A^x  +  Biy  +  C^z  +  Z>2  =  0,  Azx  +  Bzy  +  Ca^  +  -D3  =  0  belong  to  the  same 
pencil,  are 


A{-\-  kAt  _  Bi  +  kB2 
As  Bz 


Ci  -t-  kCj 
Cz 


Di±kD2. 
Dz       ' 


or,  putting  these  fractions  equal  to  s  and  eliminating  k  and  a, 


Ci 

Di 

Ci 

Di 

C2 

Di 

= 

C2 

Dt 

Cz 

Dz 

Cz 

Dz 

A, 
A2 
Az 


Di 
D2 
Dz 


Ai 
Ao 
Az 


Ai 
A2 
Az 


By 

Cx 

B2 

C2 

Bz 

Cz 

=  0. 


(Verify  Ex.  2  by  using  these  conditions.) 


15.  Find  the  equations  of  the  faces  of  a  right  pyramid,  with  square 
base  of  side  2  a  and  with  altitude  h,  the  origin  being  taken  at  the  center 
of  the  base,  the  axis  Oz  through  the  opposite  vertex,  and  the  axes  Ox,  Oy 
parallel  to  the  sides  of  the  base. 

16.  Homogeneous  substances  passing  from  a  liquid  to  a  solid  state  tend 
to  form  crystals ;  e.g.  an  ideal  specimen  of  ammonium  alum  has  the  form 
of  a  regular  octahedron.  Find  the  equations  of  the  faces  of  such  a  crystal 
of  edge  a  if  the  origin  is  taken  at  the  center  and  the  axes  through  the 
vertices,  and  determine  the  angle  between  two  faces. 

17.  Find  the  angles  between  the  lateral  faces  of  a  right  pyramid  whose 
base  is  a  regular  hexagon  of  side  a  and  whose  altitude  is  h. 


XI,  §  232] 


THE  STRAIGHT  LINE 


219 


PART   11.     THE   STRAIGHT   LINE 

231.   Determination  of  Direction  Cosines.    Two  simulta- 
neous linear  equations  (§  227), 
(1)  Ax+By  +  Cz-{-n=0,   A'x-{-B'y-{-C'z-\-I>'=0, 

represent  a  line,  namely,  the  intersection  of  the  two  planes 
represented  by  the  two  equations  separately,  provided  the  two 
planes  are  not  parallel. 

To  obtain  the  direction  cosines  ?,  m,  n  of  this  line  observe 
that  the  line,  since  it  lies  in  each  of  the  two  planes,  is  perpen- 
dicular to  the  normal  of  each  plane.  Now,  by  §  222  the  direc- 
tion cosines  of  these  normals  are  proportional  to  A,  B,  0  and 
A' J  B\  C\  respectively.     We  have  therefore 

Al  +  Bm-\-Cn  =  0,   An  +  B'm  -h  C"n  =  0, 


whence 


l:m:n 


BO 
B'C' 


OA 
C'A' 


AB 
A'B' 


The  direction  cosines  themselves  are  then  found  by  dividing 
each  of  these  determinants  by  the  square  root  of  the  sum  of 
their  squares. 


232.   Intersecting  Lines.     The  two  lines 
A^x-{-B,y-\-  Ci^-f  A  =  0,  )  r  A^-^B^y  +  G^^  +  D^^O, 

A^x  -\-  B^y  +  C^z  4-  A'  =  0  J  ^^     1  Ao}x^B^y-\-  C^z+D^  =  0 

will  intersect  if,  and  only  if,  the  four  planes  represented  by 
these  equations  have  a  common  point.  By  §  227,  the  condition 
for  this  is 

A   A   C'l   A 

A^  B,'  C/  A 

A2      Jj2      C2     -^2 
A2      -O2      ^2     -^2 


=  0. 


220  SOLID  ANALYTIC  GEOMETRY        [XI,  §  233 

233.  Special  Forms  of  Equations.  For  many  purposes  it  is 
couvenieiit  to  represent  a  line  by  means  of  one  of  its  points 
and  its  direction  cosines,  or  by  means  of  two  of  its  points. 
Let  the  line  be  called  A. 

If  (^j  ^ij  2;i)  is  a  given  point  of  X  and  I,  m,  n  are  the  direc- 
tion cosines  of  X,  then  every  point  (a;,  y,  z)  of  A  must  satisfy 
the  relations  (§  203)  : 

In  these  equations,  I,  m,  n,  can  evidently  be  replaced  by  any 

three  numbers  proportional  to  Z,  m,  n.      Thus,  if  («2,  ya?  ^Jg)  be 

any  point  of  \  different  from  (a^,  ^u  ^j),  we  have  the  continued 

proportion 

X2  —  Xi :  y2  —  yi :  z^  —  Zi  =  l :  m  :  n; 

hence  the  equations  of  the  line  through  the  two  points  (xi ,  yi ,  Zi) 

and  (x2 ,  2/2  f  ^2)  21  re : 

(S)  x-x^  ^  y-Vi  ^  g-gj  ^ 

X^-QC^        2/2-2/1        «2-«i* 

If,  for  the  sake  of  brevity,  we  put  x^—  x^  =  a,  y2  —  yi  =  ^> 
Z2  —  Zi  =  c,  we  can  write  the  equations  of  the  line  in  the  form 

(4)  ag-a?i^2/-2/i_g-gi 

a  be* 

where  a,  b,  c,  are  proportional  to  I,  m,  n,  and  can  be  regarded  as 
the  components  of  a  vector  parallel  to  the  line. 

The  equations  (3)  also  follow  directly  by  eliminating  k  be- 
tween the  equations  of  §  200,  namely, 

(5)  x=xi+k{x2-x{),  2/=2/i+A:(2/2-2/i),  z=Zj^+k(z2-Zj). 

These  equations  which,  with  a  variable  k,  represent  any  point 
of  the  line  through  (x^,  y^,  z^  and  (ajj,  2/2?  ^2)  are  called  the 
parameter  equations  of  the  line. 


XI,  §  234] 


THE  STRAIGHT  LINE 


221 


Fig.  120 


234.  Projecting  Planes  of  a  Line.  Each  of  the  forms  (2), 
(3),  (4),  which  are  not  essentially  different,  furnishes  three 
linear  equations ;  thus  (4)  gives  : 

6  c  G  a  ah 

but  these  three  equations  are  equivalent  to  only  two,  since  from 
any  two  the  third  follows  immediately. 
The    first  of  these   equations,   which 
can  be  written  in  the  form 

represents,  since  it  does  not  contain  x 
(§  218),  a  plane  perpendicular  to  the 
plane  Oyz]  and  as  this  plane  must  con- 
tain the  line  X  it  is  the  plane  CCA 
that  projects  \  on  the  plane  Oyz  (Fig.  120).  Similarly  the  other 
two  equations  represent  the  planes  that  project  A.  on  the  co- 
ordinate planes  Ozx  and  Oxy.  Any  two  of  these  equations 
represent  the  line  X  as  the  intersection  of  two  of  these  pro- 
jecting planes. 

At  the  same  time  the  equation 

y-?/i^g-gi 

h  c 

can  be  interpreted  as  representing  a  line  in  the  plane  Oyz, 
viz.  the  intersection  of  the  projecting  plane  with  the  plane 
x  —  0.  This  line  {AC  in  Fig.  120)  is  the  projection  X^  of  X  on 
the  plane  Oyz.  As  the  other  two  equations  (4)  can  be  inter- 
preted similarly  it  appears  that  the  equations  (2),  (3),  or  (4) 
represent  the  line  A.  by  means  of  its  projections  A^,  Xy,  X^  on 
the  three  coordinate  planes,  just  as  is  done  in  descriptive 
geometry.  Any  two  of  the  projections  are  of  course  sufficient 
to  determine  the  line. 


222  SOLID  ANALYTIC  GEOMETRY         [XI,  §  235 

235.  Determination  of  Projecting  Planes.  To  reduce  the 
equations  of  a  line  A  given  in  the  form  (1)  to  the  form  (4)  we 
have  only  to  eliminate  between  the  equations  (1)  first  one  of 
the  variables  x,  y,  z,  then  another,  so  as  to  obtain  two  equa- 
tions, each  in  only  two  variables  (not  the  same  in  both). 

The  process  will  best  be  understood  from  an  example.     The 
line  being  given  as  the  intfersection  of  the  planes 
(a)  2x-Sy-{-z  +  S  =  0y 
(5)  x  +  y  +  z-2  =  0, 

eliminate  z  by  subtracting  (6)  from  (a)  and  eliminate  x  by 
subtracting  (6),  multiplied  by  2,  from  (a) ;  this  gives  the  line 
as  the  intersection  of  the  planes 

x  —  4:y-\-5  =  0, 

which  are  the  projecting  planes  parallel  to  Oz  and  Ox,  i.e.  the 
planes  that  project  the  line  on  Oxy  and  Oyz.  Solving  for  y 
and  equating  the  two  values  of  y  we  find : 

x-\-5  __y  _z  —  7 
4    ^i~35"- 

The  line  passes  therefore  through  the  point  (—5,  0,  7)  and 
has  direction  cosines  proportional  to  4,  1,  —  5,  viz. 

,4  1  5 

1  =  — ^,        m  =  — =,         n  = — . 

V42  V42  V42 

EXERCISES 

1.  Write  the  equations  of  the  line  through  the  point  (—  3, 1,  6)  whose 
direction  cosines  are  proportional  to  3,  5,  7. 

2.  Write  the  equations  of  the  line  through  the  point  (3,  2,  —  4)  whose 
direction  cosines  are  proportional  to  6,  —  1,  3. 

3.  Find  the  line  through  the  point  (a,  6,  c)  that  is  equally  inclined 
to  the  axes  of  coordinates. 


XI,  §  236]  THE  STRAIGHT  LINE  223 

4.   Find  the  lines  that  pass  through  the  following  pairs  of  points : 
(a)  (4,  -  3,  1),  (2,  3,  2),  (6)  (-  1,  2,  3),  (8,  7,  1), 

(c)  (-  2,  3,  -  4),  (0,  2,  0),  (d)  (-  1,  -  5,  -  2),  (-  3,  0,  -1), 

and  determine  the  direction  cosines  of  each  of  these  lines. 

6.  Find  the  traces  of  the  plane  2aj  —  3?/  —  4^  =  6  in  the  coordinate 
planes. 

6.  Write  the  equations  of  the\me2x—Sy-\-6  z—6=0,x—y-\-2z  —  S=0 
in  the  form  (4)  and  determine  the  direction  cosines. 

7.  Put  the  line  4x  — Sy  — 6  =  0^x  — y  —  z  — 4:  =  0  in  the  form  (4) 
and  determine  the  direction  cosines. 

8.  Find  the  line  through  the  point  (2,  1,  —  3)  that  is  parallel  to  the 
line  2x-Sy  +  4z  —  6  =  0,  6x  +  y-2z  —  S  =  0. 

9.  What  are  the  projections  of  the  line  6x  — Sy  —  lz  — 10  =  0, 
x  +  y  —  Sz+6  =  0  on  the  coordinate  planes  ?    - 

10.  Obtain  the  equations  of  the  line  through  two  given  points  by 
equating  the  values  of  k  obtained  from  §  200. 

11.  By  §  222,  the  direction  cosines  of  any  line  are  proportional  to  the 
coeflBcients  of  x,  y,  and  z  in  the  equation  of  a  plane  perpendicular  to  the 
line.  Find  a  line  through  the  point  (3,  5,  8)  that  is  perpendicular  to  the 
plane  2x-\-y  +  Sz=:b. 

236.  Angle  between  Two  Lines.  The  cosine  of  the  angle  ^  be- 
tween two  lines  whose  direction  cosines  are  Zi ,  mi ,  wi  and  Z2  >  wi2 ,  n2  is, 

by  §  204, 

cos  \j/  =  hh  +  Wim2  +  W1W2 . 

Hence  if  the  lines  are  given  in  the  form  (4) ,  say 

x-xi^y  —  yi_z  -  zi      x  -Xi  _y  —  yi  _z  —  Zj 


we  have 

COS^    : 

hi 

ci            ai           hi            C2 
aiGi  4-  hihi  +  C1C2 

±V^ 

■  +  5l2  +  Ci2  . 

±Va22  +  &2^  +  C22 

If  the  lines  are  parallel^ 

then 

ai_&i. 
ai     hi 

—        > 
Ci 

if  they  are  perpendicular, 

then 

and  mce  versa. 

didi  +  &1&2  + 

C1C2  =  0  ; 

224 


SOLID  ANALYTIC  GEOMETRY         [XI,  §  237 


Let  the  line  and  plane 


plana 


237.   Angle  between  Line  and  Plane. 

be  given  by  the  equations 

x  —  x\_y—  y\_z  —  zi 
a  b  c     ' 

Ax-{-By+Cz  +  D  =  0. 

The  plane  of  Fig,  121  represents  the  plane 
through  the  given  line  perpendicular  to  the  given 
plane.     The  angle  /3  between  the  given  line  and 
plane  is  the  complement  of  the  angle  a  between  the  line  and  any  perpen- 
dicular PN  to  the  plane.     Hence 

8in^=  aA  +  bB  +  cC 


Fig.  121 


±  Va'^  +  6'-«  +  c2 .  ±  y/A^  +  52  +  C^ 
The  (necessary  and  sufficient)  condition  for  parallelism  of  line  and 

plane  is 

aA -\- bB -\- cC  =  0  i 

the  condition  of  perpendicularity  is 

a  _b_± 
A~  B     C 

238.  Line  and  Plane  Perpendicular  at  Given  Point.    If  the 

plane  Ax -^  By -^  Cz  +  D  =  0 

passes  through  the  point  Pi(xi ,  yi ,  «i),  we  must  have 
Axi  +  Byi  -{-  Czi -\-  D  =  0. 
Subtracting  from  the  preceding  equation,  we  have  as  the  equation  of 
any  plane  through  the  point  Pi(xi ,  yi,  zi)  : 

A{x  -  xi)  +  B{y  -  yi)  +  C(z  -  zi)  =  0. 
The  equations  of  any  line  through  the  same  point  are 

x  —  xi  __  y  —  y\  _  z  —zx 
a  b  c 

If  this  line  is  perpendicular  to  the  plane,  we  must  have  (§  237)  :  a/ A  = 
b/B  =  c/C.    Hence  the  equations 

x  —  xi_y—y\_z  —  zi 
A     ~     B     ~     C 

represent  the  line  through  Pi(a;i,  yi,  zi)    perpendicular  to  the  plane 

A(x  -  xi)  +  B{y  -  yi)  +  C{z  -  zi)  =  0. 


XI,  §  240] 


THE  STRAIGHT  LINE 


225 


239.  Distance  of  a  Point  from  a  Line,    if  the  equations  of 

the  line  X  are  given  in  the  form 

z  —  Z\ 


I 


y-yi 

m 


where  (cci ,  yi ,  zi)  is  a  point  Pi  of  X  (Fig. 
122),  the  distance  d  =  QP2  of  the  point 
P2ix2,  2/2,  Zi)  from  X  can  be  found  from 
the  right-angled  triangle  P1QP2  which  gives 

d^  =  P1P2' -  PlQ^ 
by  observing  that 

P1P22  =  (X2  -  Xi)2  +  (y2  -yiy  +  (Z2  -  ZiY, 

while  PiQ  is  the  projection  of  P1P2  on  X.  This  projection  is  found 
(§  199)  as  the  sum  of  the  projections  of  the  components  X2  —  xi,  2/2  —  2/1, 
Z2  —  z\  of  P1P2  on  X : 

PiQ  =  1{X2  -  xi)  +  w(2/2  -  y\)  +  n{Z2  -  Zl). 
Hence 

cP=(^X2-Xiy+  (2/2-2/i)H(^2-2i)2-[Z(x2-xi)4-w(2/2-2/i)  +n(z2-z{)Y. 
240.   Shortest  Distance    between   Two  Lines.     Two  lines 

Xi ,  X2  whose  equations  are  given  in  the  form 

x  —  xi_y  —  yi_z  —  zi     x~  X2  _y  —  y2  _z  -  Z2 


TOl 


Wl 


m2 


W2 


xoill  intersect  if  their  directions  {l\ ,  «ii ,  wi),  (Z2,  WI2 ,  W2),  and  the  direc- 
tion of  the  line  joining  the  points  (xi ,  2/1 ,  ^1),  (^2 ,  2/2 »  ^2)  are  complanar 
(§207),  i.e.  if 

X2  —  Xi     2/2  -  2/1     «2  -  «1 
Zl  wi  n\        =  0. 

Z2  W2  W2 

If  the  lines  Xi ,  X2  do  not  intersect,  their  shortest  distance  d  is  the  dis- 
tance of  P2(X2,  2/2,  ^2)  from  the  plane  through  Xi  parallel  to  X2.  As  this 
plane  contains  the  directions  of  Xi  and  X2 ,  the  direction  cosines  of  its  nor- 
mal are  (§  206)  proportional  to 


mi     wi 

m    h 

h    mi 

m2      W2 

112    h 

' 

h    mz 

226 


SOLID  ANALYTIC  GEOMETRY        [XI,  §  240 


and  as  it  passes  through  Pi  (xi ,  yi ,  zi)  its  equation  can  be  written  in  the 
form 

x  —  xi    y  —  yi    z  —  zx 

l\  mi  m      =  0. 

I2  Wl2  712 

Hence  the  shortest  distance  of  the  lines  \\ ,  X2  is : 

Xi  -  xi   2^2  -  yi   Zi  -zi 

l\  TWi  n\ 

h WI2  rii 


V 


r/ii     n\ 

W2      W2 


h    n»i 
h    n»2 


As  the  denominator  of  this  expression  is  equal  to  sin^  (§  206),  we  have 
d  sin  ^  = 


X2  —  xi    yi  —  yi    Zi  -  zi 

h  mi  ni 

h  rrii  m 


EXERCISES 

1.   Find  the  cosine  of  the  angle  between  the  lines 

x-S_y-5^z  +  l  ^^^  x+l^y-3^g  +  3 
2  3  4  -1  2  3     * 

'  2.   Find    the    angle     between    the     lines    3x  —  2y  +  4«— 1  =  0, 
2x  +  y-30  +  10  =  0,  a,nd  x  +  y  +  z  =  6,  2x-hSy  -5z  =  S. 

3.  Find  the    angle  between  the  lines  that  pass  through  the  points 
(4,  2,  6),  (-  2,  4,  3)  and  (-  1,  4,  2),  (4,  -  2,  -  6). 

4.  Find  the  angle  between  the  line 

x  +  l_y  —  2__g  +  10 

3  -6  3 

and  a  perpendicular  to  the  plane  4x  —  3y  —  22  =  8. 

5.  In  what  ratio   does  the  plane   3x  —  4y  +  60  —  8  =  0  divide    the 
segment  drawn  from  the  origin  to  the  point  (10,  —  8,  4). 

6.  Find  the  plane  through  the  point  (2,  —  1,  3)  perpendicular  to  the 

line 

x—S_y+2_z—7 

4  3  -1  * 


XI,  §  240]  THE  STRAIGHT  LINE  227 

7.  Find  the  plane  that  is  perpendicular  to  the  line  4x  +  y  —  z=6, 
'dx  +  iy-{-Sz  +  10  =  0  and  passes  through  the  point  (4,  —  1,3). 

8.  Find  the  plane  through  the  origin  perpendicular  to  the  line 

5x-2y  +  z=6,   Sx  +  y-4z  =  S. 

9.  Find  the  plane  through  the  point  (4,  —  3,  1)  perpendicular  to  the 
line  joining  the  points  (3,  1,  —  6),  (—  2,  4,  7). 

10.  Find  the  line  through  the  point  (2,  —  1,  4)  perpendicular  to  the 
plane  ic  —  2^  +  42  =  6. 

11.  Show  that  the  lines  x/S  =  y/  -- 1  =  z/-2  and  x/i  =  y/6  =  z/3  are 
perpendicular. 

12.  Show  that  the  lines 

^L=Ll  =  y±l  =  ^:zl  and  a;-2^y-3^    z 
1-2  3  _2  4  -6 

are  parallel. 

13.  Find  the  angle  between  the  line  Sx  —  2y  —  z  =  4,  4x  +  3y  —  3^  =  6 
and  the  plane  x  -i-y  -\-  z  =  S. 

14.  Find  the  lines  bisecting  the  angles  between  the  lines 

x  —  a_y  -  &_^  —  c  ^j^j  x  —  a_y  —  b  _z  —  c 

15.  Find  the  plane  perpendicular  to  the  plane  Sx  —  iy  —  z  =  6  and 
passing  through  the  points  (1,  3,  —  2),  (2,  1,  4) . 

16.  Find  the  plane  through  the  point  (3,  —  1,  2)  perpendicular  to  the 
line  2  a;  — 3!/ —  42;  =  7,   ic+y— 2^=4. 

17.  Find  the  plane  through  the  point  (a,  6,  c)  perpendicular  to  the 
line  Aix  +  Biy  +  Ciz  +  Di  =  0,   Azx  +  B^y  +  dz  +  D2  =  0. 

18.  Find  the  projection  of  the  vector  from  (3,  4,  5)  to  (2,  —  1, 4)  on  the 
line  that  makes  equal  angles  with  the  axes  ;  and  on  the  plane 

2x-3y  +  45!=6. 

19.  Find  the  distances  from  the  following  lines  to  the  points  indicated : 

(6)  2a;  +  y-0  =  6,  a;- y +  4^  =  8,  (3,  1,4); 

(c)  2x  +  3?/  +  60  =  l,  3x-6y  +  32=0,  (4,  1,  -2). 


228  SOLID  ANALYTIC  GEOMETRY        [XI,  §  240 

20.   Show  that  the  equation  of  the  plane  determined  by  the  line 


x-xi     y-yi_z-zi 

a            b            c 

d  the  point  P2(a;2,  2/2 ,  Z2)  can  be  written  in  the  form 

X  -xi    2/  -2/1    z  -zi 

X2  —  Xi      2/2  —  Vl      Z2  —  Zi 

=  0. 

a             b             c 

21.   Find  the  plane  determined  by  the  intersecting  lines 

a:-3_y-6_0  +  l  .^.  x  -  S  _y  -  5  _z  +  1 
4             3             2                   12            3 

22.   Find  the  plane  determined  by  the  line 

x-xi_y-yi_z~zi 

a  b  c 

and  its  parallel  through  the  point  P2{x2 , 2/2 ,  ^2)- 

23.  Given  two  non-intersecting  lines 

x  —  xi  __  y  —  yi  _  z-  zx     x  —  X2  _  y  -y2  _  z  —  Zj , 
a\  b\  c\  a2  62  C2 

find  the  plane  passing  through  the  first  line  and  a  parallel  to  the  second; 
and  the  plane  passing  through  the  second  line  and  a  parallel  to  the  first. 

24.  What  is  the  condition  that  the  two  lines  of  Ex.  23  intersect  ? 

26.  Find  the  distance  from  the  diagonal  of  a  cube  to  a  vertex  not  on 
the  diagonal. 

26.  Find  the  distance  between  the  lines  given  in  Ex.  23. 

27.  Show  that  the  locus  of  the  points  whose  distances  from  two  fixed 
planes  are  in  constant  ratio  is  a  plane. 

28.  Show  that  the  plane  (m  —  n)x  +(n  —  l)y  +(l  —  m)z  =  0  contains 
the  line  x/l  =  y/m  =  z/n  and  is  perpendicular  to  the  plane  determined  by 
the  lines  x/m  =  y/n  =  z/l  and  x/n  =  y/l  =  z/m. 


CHAPTER  XII 

THE    SPHERE 

241.  Spheres.  A  sphere  is  defined  as  the  locus  of  all  those 
points  that  have  the  same  distance  from  a  fixed  point. 

Let  C{h,  j,  h)  denote  the  center,  and  r  the  radius,  of  a  sphere ; 
the  necessary  and  sufficient  condition  that  any  point  P{Xj  ?/,  z) 
has  the  distance  r  from  C{U,  j,  k)  is 

(1)  (^  -  7^)2  +  (y  -jY  -Y{z-  IcY  =  r^. 

This   then   is   the   cartesian   equation   of  the   sphere  of  center 
C(h,  j,  k)  arid  radius  r. 

If  the  center  of  the  sphere  lies  in  the  plane  Oosy,  the  equa- 
tion becomes 

(x-hy  +  {y-jy  +  z'=r\ 

If  the  center  lies  on  the  axis  Ox,  the  equation  is 

(x-hy-\-y^-\-z^  =  r\ 

The  equation  of  a  sphere  about  the  origin  as  center  is : 

242.  Expanded  Form.  Expanding  the  squares  in  the  equa- 
tion (1),  we  find  the  equation  of  the  sphere  in  the  form 

x'^-\.y^-{.z'^-2hx-  2jy  -2kz-^h'' +f-\-k'^-r^  =  0. 

This  is  an  equation  of  the  second  degree  in  x,  y,  z ;  but  it  is  of 
a  particular  form. 

The  general  equation  of  the  second  degree  in  x,  y,  z  is 

(2)  Ax"^  +  By^  -\-Cz^^2  Dyz  +  2Ezx-[-2  Fxy 

-i-2Gx-\-2Hy-{-2Iz-i-J=0', 
229 


230  SOLID  ANALYTIC  GEOMETRY      [XII,  §  242 

i.e.  it  contains  a  constant  term  J;  three  terms  of  the  first 
degree,  one  in  x,  one  in  y,  and  one  in  z ;  and  six  terms  of  the 
second  degree,  one  each  in  x^,  y^,  z^j  yz,  zx,  and  on/. 

If  in  (2)  we  have  D  =  E  =  F=  0,  A  =  B=C^O,\t  reduces, 
upon  division  by  A,  to  the  form 

a;2-f.7/2  +  2;2-f_-x+— 2/-I-— Z  +  — =  0, 

which  agrees  with  the  above  form  of  the  equation  of  a  sphere, 
apart  from  the  notation  for  the  coefficients. 

243.  Determination  of  Center  and  Radius.     To  determine 
the  locus  represented  by  the  equation 
(3)        Ax^  -\-  Ay"  +  Az^  ■\-2  Gx^-2  Hy  -\-2  Iz  -^r  J^Q, 
where  A^  G,  II,  /,  J  are  any  real  numbers  while  ^  =^  0,  we 
divide  by  A  and  complete  the  squares  in  x,  y,  z;  this  gives 

The  left  side  represents  the  square  of  the  distance  of  the  point 
{x,  y,  z)  from  the  point  (—  G/A,  —H/A,  —  I/A)]  the  right 
side  is  constant.  Hence,  if  the  right  side  is  positive,  the  equa- 
tion represents  the  sphere  whose  center  has  the  coordinates 
(—  G/A,  —  H/A,  —  I/A),  and  whose  radius  is 


r  =  -WG^  +  H^-\-P-AJ. 
A 

If,  however,  G^  +  H^  +  I^<  AJ,  the  equation  is  not  satisfied  by 

any  point  with  real  coordinates.     If   G^  -|-  H^  -{-/*  =  AJ,  the 

equation  is   satisfied   only  by  the  coordinates  of  the   point 

(-G/A,-H/A,-I/A). 

Thus  the  equation  of  the  second  degree 

Ax^  +  By^-j-Cz^  +  2  Dyz  -\-2Ezx-{-2  Fxy 

+  2Gx-{-2Hy  +  2Iz-i-J=0, 

represents  a  sphere  if,  and  only  if, 

A=B=C^O,    D=:E  =  F=0,     G^  +  IP-\-P>AJ. 


XII,  §  244]  THE  SPHERE  231 

244.  Essential  Constants.  The  equation  (1)  of  the  sphere 
contains  four  constants  :  li,  j^  k,  r.  The  equation  (2)  contains 
five  constants  of  which,  however,  only  four  are  essential  since 
we  can  divide  out  by  one  of  these  constants.  Thus  dividing 
by  A  and  putting  2  0/A  =  a,  2  H/A  =  6,  2  I/A  =  c,  J/ A  =  d, 
the  general  equation  (2)  assumes  the  form 

x^  +  y'^  -\-  z^  +  ax  -[-hy  -\-  cz  -\-  d  =  0, 
with  only  the  four  essential  constants  a,  6,  c,  d. 

This  fact  corresponds  to  the  possibility  of  determining  a 
sphere  geometrically,  in  a  variety  of  ways,  by  four  conditions. 

EXERCISES 

1.  Find  the  spheres  with  the  following  points  as  centers  and  with  the 
indicated  radii : 

(a)   (4,  -1,2),  4;    (6)   (0,0,  4),  4;  (c)   (2,-2,  1),  3 ;  {d)   (3,  4,  1),  7. 

2.  Find  the  following  spheres : 

(«)  with  the  points  (4,  2,  1)  and  (3,  —  7,  4)  as  ends  of  a  diameter ; 

(6)  tangent  to  the  coordinate  planes  and  of  radius  a  ; 

(c)  with  center  at  the  point  (4,  1,  5)  and  passing  through  (8,  3,  —  5). 

3.  Find  the  centers  and  the  radii  of  the  following  spheres  : 
(a)  a;2  +  2/2  +-^2 _3a;  +  5y_6^  +  2  =  0. 

(6)  a:2  +  y2  _|.  2;2  -  2  6a;  +  2  c;?  -  62  _  c2  =  0. 
(c)   2  x2  +  2  1/2  +  2  2;2  -^  3  a;  -  y  +  5  0  -  11  =  0. 
{d)  x^-{-y'^  +  z^-x-y  -z  =  Q. 

4.  Show  that  the  equation  A{x'^  ■\- y"^  +  z"^)  +  2  Ox  +  2  Hy  -^  2  Iz  +  J 
=  0,  in  which  J  is  variable,  represents  a  family  of  concentric  spheres. 

5.  Find  the  spheres  that  pass  through  the  following  points  : 
(a)   (1,  1,  1),  (3,  -  1,  4),  (-  1,  2,  1),  (0,  1,  0). 

(6)   (0,  0,  0),  (a,  0,  0),  (0,  6,  0),  (0,  0,  c). 

(c)  (0,  0,  0),  (-  1,  1,  0),  (1,  0,  2),  (0,  1,  -  1). 

(d)  (0,  0,  0),  (0,  0,  4),  (3,  3,  3),  (0,  4,  0). 

6.  Find  the  center  and  radius  of  the  sphere  that  is  the  locus  of  the 
points  three  times  as  far  from  the  point  (a,  6,  c)  as  from  the  origin. 

7.  Show  that  the  locus  of  the  points,  the  ratio  of  whose  distances  from 
two  given  points  is  constant,  is  a  sphere  except  when  the  ratio  is  unity. 


232  SOLID  ANALYTIC  GEOMETRY       [XII,  §  244 

8.  Find  the  positions  of  tlie  following  points  relative  to  the  sphere 
x^  +  y^  +  z'^-4x-\-^y-2z  =  0]  (a)  the  origin,  (6)  (2,  -2,  1), 
(c)   (1,1,1),  (d)  (3,  -2,1). 

9.  Find  the  positions  of  the  following  planes  relative  to  the  sphere 

(a)4x-\-2y  +  z-\-2  =  0,  (b)8x-y-iz-\-5  =  0. 

10.  Find  the  positions  of  the  following  lines  relative  to  the  sphere  of 
Ex.  9:  (a)2x-y  +  2z  +  7=0,    Zx-     y-z-lOz=0. 

(6)3x  +  8y  +  5r-9=0,       a;-8y+«+ll  =  0. 

245.  Equations  of  a  Circle.  In  solid  analytic  geometry  a 
curve  is  represented,  by  two  simultaneous  equations  (§  221), 
that  is,  by  the  equations  of  any  two  surfaces  intersecting  in 
the  curve.  Thus  two  linear  equations  represent  together  the 
line  of  intersection  of  the  two  planes  represented  by  the  two 
equations  taken  separately  (§§  233,  237). 

A  linear  equation  together  with  the  equation  of  a  sphere, 

^  ^  x^ -\- y^ -\- z^  -{- ax -j- by  +  cz -{- d  =  0, 

represents  the  locus  of  all  those  points,  and  only  those  points, 
which  the  plane  and  sphere  have  in  common.  Thus,  if  the 
plane  intersects  the  sphere,  these  simultaneous  equations  rep- 
resent the  circle  in  which  the  plane  cuts  the  sphere;  if  the 
plane  is  tangent  to  the  sphere,  the  equations  represent  the 
point  of  contact;  if  the  plane  does  not  intersect  or  touch 
the  sphere,  the  equations  are  not  satisfied  simultaneously  by 
any  real  point. 

246.  Sections  Perpendicular  to  Axes.  Projecting  Cylinders. 

In  particular,  the  simultaneous  equations 

(6)  z  —  k,         a^  -\- y^  -\- z^  =  r^ 

represent,  if  A:  <  r,  a  circle   about  the  axis  Oz   (i.e.  a  circle 

whose  center  lies  on  Oz  and  whose  plane  is  perpendicular  to 

Oz).     If  the  value  of  z  obtained  from  the  linear  equation  be 


XII,  §  247]  THE  SPHERE  233 

substituted  in  the  equation  of  the  sphere,  we  obtain  an  equation 
in  X  and  ?/,  oc^  -{- 1/  —  r^  —  Tc^,  which  represents  (since  z  is 
arbitrary)  the  circular  cylinder,  about  Oz  as  axis,  which  pro- 
jects the  circle  (5)  on  the  plane  Oxy.  Interpreted  in  the  plane 
Oxy^  i.e.  taken  together  with  z  =  0,  this  equation  represents 
the  projection  of  the  circle  (5)  on  the  plane  Oxy. 

Similarly  if  we  eliminate  x  ov  y  or  2  between  the  equations 
(4),  we  obtain  an  equation  in  y  and  z,  z  and  x,  or  x  and  y,  rep- 
resenting the  cylinder  that  projects  the  circle  (4)  on  the  plane 
OyZy  Ozx,  or  Oxy,  respectively. 

247.  Tangent  Plane.  The  tangent  plane  to  a  sphere  at  any 
point  Pi  of  the  sphere  is  the  plane  through  Pj,  at  right  angles 
to  the  radius  through  Pj . 

¥oY  a  sphere  whose  center  is  at  the  origin,  a;^  +  2/^  -f  2^  _  ^^ 
the  equation  of  the  tangent  plane  at  Pi(x^,  yi,  z^)  is  found  by 
observing  that  its  distance  from  the  origin  is  r  and  that  the 
direction  cosines  of  its  normal  are  those  of  OPi,  viz.  Xi/r, 
yi/r,  Zi/r.  Hence  the  equation 
(6)  x^x  +  y{y  +  212;  =  r\ 

If  the  equation  of  the  sphere  is  given  in  the  general  form 
^(a;2  +2/2+  22)4.  2  Gx  +  2  Hy  +2Iz  -\-J=  0, 
we  obtain  by  transforming  to  parallel  axes  through  the  center 
the  equation 

the  tangent  plane  at  Fi(xi,  yi,  Zi)  then  is 

.  x,x  +  2/12/  +  ^i2  =  —  +  —  +  —  -  -. 
Transforming  back  to  the  original  axes,  we  have : 

A^      A^     A^     a' 


234  SOLID  ANALYTIC  GEOMETRY      [XII,  §  247 

Multiplying  out  and  rearranging,  we  find  that  the  equation  of 
the  tangent  plane  to  the  sphere 

Aix"  +  y""  +  z")  +  2  Gx  +  2  Hy  +  2  Iz  +  J  =  0 
at  the  point  Pi(a^,  yi,  Zi)  is 

A{x^xJtyiy^-Ziz)  +  0(x^^x)  4-fi-(2/i+y)4- 7(21+2)+  J=  0. 

248.   Intersection  of  Line  and  Sphere.     The  intersections 
of  a  sphere  about  the  origin, 

x^  -^y^  +  z^  =  r\ 
with  a  line  determined  by  two  of  its  points  Pi(xi,  yi,  Zi)  and 
PiiXi,  2/2>  ^2)}  a-iid  given  in  the  parameter  form  [(6),  §  239] 

x  =  x^  +  k{x2-x{),  y  =  yi  +  k(y2-yi)j  2  =  %  +  ^(za  -  Zi), 

are  found  by  substituting  these  values  of  a:,  y,  z  in  the  equation 

of  the  sphere  and  solving  the  resulting  quadratic  equation  in  k : 

Ix,  +  k(x,  -  x{)Y  +  [2/1  +  k(y,  -  2/0]^  +  [^1  +  k{z,  -  z,)Y  =  r», 

which  takes  the  form 

l(x,  -  x,y  +  (y,  -  y,y  +  (z,  -  z,y^ k'  +  2  [x,{x,  -  x,)+  y,(y,  -  y,) 
+  Zi  (22  -  Zi)]  k  +  W  +  2/1'  +  z,'  -  r2)  =  0. 
The  line  P1P2  will  intersect  the  sphere  in 
two  different  points,  be  tangent  to  the 
sphere,  or  not  meet  it  at  all,  according  as  ^C^f^ ^^ 

the  roots  of  this  equation  in  k  are  real  and    ^x^ 
different,  real  and  equal,  or  imaginary ;  i.e. 
according  as  ^^^'  ^^ 

lx,{x,-x,)+y,(y,-y,)+z,(z,-z,)J-(P{x,^-^y,^+z,^)+(Pr''^0, 

where  d  denotes  the  distance  of  the  points  Pi  and  Pj-     Divid- 
ing by  cP,  we  can  write  this  condition  in  the  form 

where  by  §  239  the  quantity  in  square  brackets  is  the  square 
of  the  distance  S  from  the  line  P^Pz  to  the  origin  0  (Fig.  123). 


XII,  §  249]  THE  SPHERE  235 

Our  condition  means  therefore  that  the  line  P^P^  meets  the 
sphere  in  two  different  points,  touches  it,  or  does  not  meet  it 
at  all  according  as  r  >  8,  r  =  8,  r  <  8,  which  is  obvious  geomet- 
rically. 

249.   Tangent  Cone.     The  condition  for  the  line  P^P^  to  be 

tangent  to  the  sphere  is  (§  244)  : 

W+  yi'-hzi'-r%)i{-x,y  H-  (y,  -  y,y  +  (22  -  z^yy 

To  give  this  expression  a  more  symmetric  form,  let  us  put,  to 

abbreviate, 

^1^2  +  2/i2/2  +  2=122  =  P,       ^i  +  yi  +  21^  =  gi,       x^  -h  2/2'  -f  z^  =  ^2, 

so  that  the  condition  is 

i.e.  p^-2  r^p  =  q^q^  -  r^q^  -  r% ; 

adding  r*  in  both  members,  we  have 

i.e. 

(x,x,  +  2/12/2  +  21^2  -  r^  =  (o^i^  +  2/1'  +  2i^  -  r')(x2'  +  2/2'  +  z,^  -  r»). 

Now  keeping  the  sphere  and  the  point  Pj  fixed,  let  Pj  vary 
subject  only  to  this  condition,  i.e.  to  the 
condition  that  P1P2  shall  be  tangent  to 
the  sphere;  the  point  Pj,  which  we  shall 
now  call  P(x,  y,  z)  is  then  any  point  of 
the  cone  of  vertex  P^  tangent  to  the  sphere. 
Hence  the  equation  of  the  cone  of  vertex 
Pi  (^1 )  2/1 J  2i)  tangent  to  the  sphere  x^-\-y'^-\-  z^ 

(a^i'  +  2/1'  +  z^  -  r2)(a^  +  y2  +  2;^  _  ^) ^ (^^^  ^  2/i2/  ^-z^z  -  rj. 

If,  in  particular,  the  point  Pi  is  taken  on  the  sphere  so  that 
x^  +  yx  +  z^  —  r\  the  equation  of  the  tangent  cone  reduces  to 
the  form  x^x  +  y^y  +  z^z  =  r^,  which  represents  the  tangent 
plane  at  Pi. 


236  SOLID  ANALYTIC  GEOMETRY       [XII,  §250 

250.  Inversion.  A  sphere  of  center  0  and  radius  a  being  given, 
we  can  find  to  every  point  P  of  space  (excepting  0)  one  and  only  one 
point  P'  on  OP  (produced  if  necessary)  such  that  OP  ■  OP'  =  a^.  The 
points  P,  P'  are  said  to  be  inverse  to  each  other  with  respect  to  the 
sphere  (compare  §  56). 

Taking  rectangular  axes  through  O,  we  find  as  the  relations  between 
the  coordinates  of  the  two  inverse  points  P(x,  y,  z)  and  P'(x',  y',  z')  if 
we  put  OP  =  r  =  Vx^  +  y^  +  z'\   OP'  =  r'  =  Vx'=^  +  y'-^  +  z'^  : 

X  _y'  _z'  _r'  _rr'  _a'^  , 
X      y      z      r       r^      r^  ' 

hence    z' -        "^"^  v'- ^ z' - — • 

hence    x-^^^^^^^^,         ^'-^2  +  ^2  +  ^2'        '-^^^y2^^2' 

and  similarly 

a^x'  ..  _  aY  „  _  a^z' 


X  = 


x'-i  ^  y'i  +  z'i'      "      x'2  +  y'i  +  z'^'  x'-^  +  y'^  +  z'^ 

These  equations  enable  us  to  find  to  any  surface  whose  equation  is  given 
the  equation  of  the  inverse  surface,  by  simply  substituting  for  x,  y,  z 
their  values. 

Thus  it  can  be  shown,  that  by  inversion  every  sphere  is  transformed 
into  a  sphere  or  a  plane.  The  proof  is  similar  to  the  corresponding  propo- 
sition in  plane  analytic  geometry  (§57)  and  is  left  as  an  exercise. 

EXERCISES 

1.  Find  the  radius  of  the  circle  which  is  the  intersection :  (a)  of  the 
plane  y  =  6  with  the  sphere  x"^  -\-  y^  -\-  z^  —  6y  =  0  ]  (6)  of  the  plane 
2x  —  Sy  +  z-2  =  0  with  the  sphere  x"^ +  y^  +  z^ -6x +  2y  -  16  =  0. 

2.  A  line  perpendicular  to  the  plane  of  a  circle  through  its  center  is 
called  the  axis  of  the  circle.  Find  the  circle  :  (a)  which  lies  in  the  plane 
«  =  4,  has  a  radius  3  and  Oz  as  axis ;  (6)  which  lies  in  the  plane  y  =  6, 
has  a  radius  2  and  the  line  x  —  3  =  0,  0  —  4=0as  axis. 

3.  Find  the  circles  of  radius  3  on  the  sphere  of  radius  4  about  the 
origin  whose  common  axis  is  equally  inclined  to  the  coordinate  axes. 

4.  Does  the  Une  joining  the  points  (2,-1,-6),  (-1,  2,  3)  intersect 
the  sphere  x^  -\-y^  +  z^  =  10?    Find  the  points  of  intersection. 


XII,  §251]  THE  SPHERE  237 

5.  Find  the  planes  tangent  to  the  following  spheres  at  the  given 
points  :     (a)  a;2  +  2/2  +  ^2  _  3  ^  _  5  ^  _  2  =  0,  at  (2,  -  1,  3)  ; 

(&)  x2  +  y2  ^  02  ^  2  X  -  6  ?/  +  2  -1  =  0,  at  (0,  1,  -  3)  ; 

(c)  3ix^  +  y^-\-z^)-5x  +  2y -z  =  0,  at  the  origin; 

(d)  x^  -^  y^  +  z'^—  ax  -  bij  -  cz  -  0,  at  (a,  b,  c). 

6.  Find  the  tangent  cone  :  (a)  from  (4,  1,  —  2)  to  a;2  +  y2  ^  ^2  =  9 ; 
(6)  from  (2  a,  0,  0)  to  x^ +  y^  +  z"^  =  a^  ;  (c)  from  (4,  4,  4)  to  a;2  +  y2 
+  ^^2  =  16 ;  (d)  from  (1,  -  5,  3)  to  x'^  +  y^  +  z^  =  9. 

7.  Find  the  cone  with  vertex  at  the  origin  tangent  to  the  sphere 
(x-2  a)2+  y'i  +  z2  =  a\ 

8.  Show  that,  by  inversion  with  respect  to  the  sphere  x^  +  y'^  -^  z^  =  a^, 
every  plane  (except  one  through  the  center)  is  transformed  into  a  sphere 
passing  through  the  origin. 

9.  With  respect  to  the  sphere  x^  -i-  y^  +  z^  =  25,  find  the  surfaces  in- 
verse to  (a)  x  =  6,  (6)  x-y  =  0,  (c)  4  (a;2  +  y2  4.  ^2)  _  20  a;  —  25  =  0. 

10.  Show  that  by  inversion  with  respect  to  the  sphere  x'^ -\- y^ -\-  z^  =  a^ 
every  line  through  the  origin  is  transformed  into  itself. 

11.  With  respect  to  the  sphere  x^  -\-y^  -\-  z^  =  a^^  find  the  surface  in- 
verse to  the  plane  tangent  at  the  point  Pi  {xi  ,^yi ,  zi). 

12.  Show  that  all  spheres  with  center  at  the  center  of  inversion  are 
transformed  into  concentric  spheres  by  inversion. 

13.  What  is  the  curve  inverse  to  the  circle  x^  +  y^  +  z^  =  25,  0  =  4, 
with  respect  to  the  sphere  x^  +  y^  -^  z^  =  16? 

251.  Poles  and  Polars.  Let  P  and  P'  be  inverse  points  with 
respect  to  a  given  sphere  ;  then  the  plane  w  through  P',  at  right  angles  to 
OP  ( 0  being  the  center  of  the  sphere) ,  is  called  the  polar  plane  of  the 
point  P,  and  P  is  called  the  pole  of  the  plane  tt,  with  respect  to  the 
sphere. 

With  respect  to  a  sphere  of  radius  a,  with  center  at  the  origin^ 
x2  +  y2  _|_  2-2  =  a2, 

the  equation  of  the  polar  plane  of  any  point  Pi(iCi,  2/1,  z{)  is  readily 
found  by  observing  that  its  distance  from  the  origin  is  a2/ri,  and  that  the 


238  SOLID  ANALYTIC  GEOMETRY       [XII,  §251 

direction  cosines  of  its  normal  are  equal  to  Xi/ri,  yi/n,  ^i/n,  where 
n^  =  xi^  -\-  y\^  +  z^ ;  the  equation  is  therefore 

xix  +  Viy  +  ziz  =  a^- 
If,  in  particular,  the  point  Pi  lies  on  the  sphere,  this  equation,  by  §  254 
(6),  represents  the  tangent  plane  at  Pi.     Hence  the  polar  plane  of  any 
point  of  the  sphere  is  the  tangent  plane  at  that  point ;  this  also  follows 
from  the  definition  of  the  polar  plane. 

262.  With  respect  to  the  same  sphere  the  polar  planes  of  any  two 
points  Pi(a;i ,  yi ,  zi)  and  P2(X2 ,  yz ,  ^2)  are 

Xix  +  yiy  +  ziz  =  a^     and     xix  +  yzy  +  z^z  =  a*. 
Now  the  condition  for  the  polar  plane  of  Pi  to  pass  through  P2  is 
v.xXi  +  yi2/2  +  z^zi  =  a?- ; 
but  this  is  also  the  condition  for  the  polar  plane  of  P2  to  pass  through  Pi. 
Hence  the  polar  planes  of  all  the  points  of  any  plane  w  (not  passing 
through  the  origin  O)  pass  through  a  common  pointy  namely,  the  pole 
of  the  plane  ir  ;  and  conversely,  the  poles  of  all  the  planes  through  a  com- 
mon point  P  lie  in  a  plane^  namely,  the  polar  plane  of  P. 

263.  The  polar  plane  of  any  point  P  of  the  line  determined  by  two 
given  points  Pi(xi ,  j/i ,  zi)  and  P2(X2 ,  2/2 ,  22)  (always  with  respect  to  the 
same  sphere  x^  -\- y^  +  z^  =  a^)  is 

Ixi  +  k{x2  -  xi)]x  +  [yi  +  k{y2  -  t/i)]y  +  \.zi-\-k{z2  -  zi)^z  =  a^. 

This  equation  can  be  written  in  the  form 

Xix  +  yiy  +  ziz  —  a^  +  — — •  (xzx  +  y2y  +  Z2Z  —  a^)  =  0, 

which  for  a  variable  k  represents  the  planes  of  the  pencil  whose  axis  is  the 
intersection  of  the  polar  planes  of  Pi  and  P2.  Hence  the  polar  planes  of 
all  the  points  of  a  line  X  pass  through  a  common  line  ;  and  conversely, 
the  poles  of  all  the  planes  of  a  pencil  lie  on  a  line. 

Two  lines  related  in  this  way  are  called  conjugate  lines  (or  conjugate 
axes,  reciprocal  polars).     Thus  the  line  P1P2 


XII,  §  256] 


THE  SPHERE 


239 


and  the  line  xix  +  yiy  +  ziz  =  a^, 

X2X  +  y2y  +  Z2Z  =  a^ 

are  conjugate  with  respect  to  the  sphere  x^  •{■  y^  -\-  s^  =  a*. 

As  the  direction  cosines  of  these  lines  are  proportional  to 

X2  —  X1,    y2-  yi,    Z2-  z\ 
and 


y\    zx 

Z\      Xx 

xi  y\ 

y2      Z2 

02  X2 

X2     y2 

respectively,  the  two  conjugate  lines  are  at  right  angles  (§236). 

254.  By  the  method  used  in  the  corresponding  problem  in  the  plane 

(§60)  it  can  be  shown  that  the  polar  plane  of  any  point  Pi(cci,  yi,  zi) 

with  respect  to  any  sphere 

A(pfi  +  2/2  +  02)  ^.  2  G^a;  +  2  ^y  +  2  70  +  J'=  0 
is 

A{xix  +  y\y  +  010)  +  G(xi  +  x)  +  H{yi  +  y)  +  /(01  4-  0)  +  J'=  0. 

255.  Power  of  a  Point,     if  in  the  left-hand  member  of  the  equation 

of  the  sphere 

(X-  /i)2  +  {y  -  j)2  +  (^z-kY-r^  =  0 
we  substitute  for  x,  y,  z\  the  coordinates  xi ,  yi ,  01  of  any  point  not  on 
the  sphere,  we  obtain  an  expression  (xi  —  hy  +  (yi  —  J)^+  {z\  —  ky  —  r^ 
different  from  zero  which  is  called  the  power  of  the  point  P\(x\  ^  yi,  01) 
with  respect  to  the  sphere. 

As  {xi  —  hy  +  {y\  —  j)'^  +  (01  —  A:)2  is  the  square  of  the  distance  d  be- 
tween the  point  Pi  and  the  center  C  of  the  sphere,  we  can  write  the 

power  of  Pi  briefly 

^2  -  r2 ; 

the  power  of  Pi  is  positive  or  negative  according  as  Pi  lies  outside  or 

within  the  sphere.     For  a  point  Pi  outside,  the  power  is  evidently  the 

square  of  the  length  of  a  tangent  drawn  from  Pi  to  the  sphere. 

256.  Radical  Plane,  Axis,  Center.    The  locus  of  a  point  whose 

powers  with  respect  to  the  two  spheres 

x2  +  y2  +  ^2  ^_  a-iX  +  hiy  +  ci0  +  di-O, 
x^+y^  +  z'^  +  a2x  +  b2y  +  C20  +  c?2  =  0 
are  equal  is  evidently  the  plane 

(ai  —  a2)x  +  (61  —  62)2/  +  (ci  -  02)0  +  di  —  d2  =  0, 
which  is  called  the  radical  plane  of  the  two  spheres.    It  always  exists  un- 
less the  two  spheres  are  concentric. 


240  SOLID  ANALYTIC  GEOMETRY       [XII,  §  256 

It  is  easily  proved  that  the  three  radical  planes  of  any  three  spheres 
(no  two  of  which  are  concentric)  are  planes  of  the  same  pencil  (§  228)  ; 
and  hence  that  the  locus  of  the  points  of  equal  power  with  respect  to 
three  spheres  is  a  straight  line.  This  line  is  called  the  radical  axis  of  the 
three  spheres  ;  it  exists  unless  the  centers  lie  in  a  straight  line. 

The  six  radical  planes  of  four  spheres,  taken  in  pairs,  are  in  general 
planes  of  a  sheaf  (§229).  Hence  there  is  in  general  but  one  point  of 
equal  power  with  respect  to  four  spheres.  This  point,  the  radical  center 
of  the  four  spheres,  exists  unless  the  four  centers  lie  in  a  plane. 

257.  Family  of  Spheres.    The  equation 

(x^-\-y^-\-z^+  aix  -\-biy-\-ciz-\-di)  -{-k(ix^+y^+z^-+a2X-\-b2y-\-C2Z-{-d2)  =0 

represents  a  family,  or  pencil,  of  spheres,  provided  k  =^—1.     If  the  two 

spheres 

x^  +  y^  +  z^  +  aix  +  biy  +  ciz  -|-  di  =  0, 

«2  +  y2  +  «2  ^a2pc  +  b2y  +  C2Z  +  d2  =  0 

intersect,  every  sphere  of  the  pencil  passes  through  the  common  circle  of 
these  two  spheres.  It  k  =—  1,  the  equation  represents  the  radical  plane 
of  the  two  spheres. 

EXERCISES 

1.  Find  the  radius  of  the  circle  in  which  the  polar  plane  of  the  point 
(4,  3,-1)  with  respect  to  x^-{-y^+Z'  =  16  cuts  the  sphere. 

2.  Find  the  radius  of  the  circle  in  which  the  polar  plane  of  the  point 
(6,  —  1,  2)  with  respect  to  x^  +  y^  -\-  z^  —  2x -\-  4y  =  0  cuts  the  sphere. 

3.  Show  that  the  plane  3a;  +  y  —  4^  =  19  is  tangent  to  the  sphere 
x^  +  y^  +  z^  —  2x  —  iy  —  6z—l2  =  0,  and  find  the  point  of  contact. 

4.  If  a  point  describes  the  plane 4a;  —  5y  —  3a:  =  16,  find  the  coordi- 
nates of  that  point  about  which  the  polar  plane  of  the  point  turns  with 
respect  to  the  sphere  x^  +y^  +  z'^  =  16. 

5.  If  a  point  describes  the  plane  2x-\-Sy-\-z  =  i,  find  that  point 
about  which  the  polar  plane  of  the  point  turns  with  respect  to  the  sphere 

X2  +  y2  +  2!2  =  8. 

6.  If  a  point  describes  the  line  ^  ~     =  ^         =  ^  ~     ,  find  the  equa- 

3  5-2' 

tions  of  that  line  about  which  the  polar  plane  of  the  point  turns  with 


XII,  §  257]  THE  SPHERE  241 

respect  to  the  sphere  x^  -\-y'^  +  z^  =  25.     Show  that  the  two  lines  are 
perpendicular. 

7.  If  a  point  describe  the  line  2x  -  Sy  -j-  iz  =  2,  x  +  y  -{■  z  =  3,  find 
the  equations  of  that  line  about  which  the  polar  plane  of  the  point  turns 
with  respect  to  the  sphere  x^  +  y^  +  z^  =  16.  Show  that  the  two  lines  are 
perpendicular. 

8.  Find  the  sphere  through  the  origin  that  passes  through  the  circle 
of  intersection  of  the  spheres  x"^  -\-  y'^  -\-  z^  —  3x  +  4  y  —  6z  —  S  =  0^ 
x^  -^  y^  +  z^  -  2  X  -\-  y  -  z  -  10  =  0. 

9.  Show  that  the  locus  of  a  point  whose  powers  with  respect  to  two 
given  spheres  have  a  constant  ratio  is  a  sphere  except  when  the  ratio  is 
unity. 

10.  Show  that  the  radical  plane  of  two  spheres  is  perpendicular  to  the 
line  joining  their'centers. 

11.  Show  that  the  radical  plane  of  two  spheres  tangent  internally  or 
externally  is  their  common  tangent  plane. 

12.  Find  the  equations  of  the  radical  axis  of  the  spheres  x^  -\-  y^-{-  z^ 
-3x-2y  -z-4:  =  0,  x^-\-y^  +  z^-\-5x~Sy-2z-8  =  0,  x'^ -\- y^ 
-\-z^-16  =  0. 

13.  Find  the  radical  center  of  the  spheres  x^ -i-y'^ -^  z^  —  6x  ■j-2y 
-  z  +  e  =  0,  x^  -\-  y^  +  z^  -  10  =  0,  x^  +  y^  +  z^  +  2x  -  Sy  +  5 z  -  6  =  0, 
^2  +  ?/2  +  02  _  2  X  +  4  ?/  -  12  =  0. 

14.  Show  that  the  three  radical  planes  of  three  spheres  are  planes  of 
the  same  pencil. 

15.  Two  spheres  are  said  to  be  orthogonal  when  their  tangent  planes 
at  every  point  of  their  circle  of  intersection  are  perpendicular.  Show 
that  the  two  spheres  x"^  -}- y^ -{-  z"^  +  a^x  +  b^y  +  c^z  +  di  =  0,  x^  4.  ^2  _|_  ^2 
+  a^x  +  h^y  +  C2«  +  0^2  =  0  are  orthogonal  when  a\ai  +  6162  +  C1C2 
=  2{di  +  d2). 

16.  Write  the  equation  of  the  cone  tangent  to  the  sphere  x^  +  y^  + 
^2  —  fi  with  vertex  (0,  0,  zi).  Divide  this  equation  by  zi^  and  let  the 
vertex  recede  indefinitely,  i.e.  let  z\  increase  indefinitely.  The  equation 
3.2  _|_  ^2  _  ^2^  thus  obtained,  represents  the  cylinder  with  axis  along  the 
axis  Oz  and  tangent  to  the  sphere  x^  +  i/^  4.  ^2  _  ,.2^ 

B 


CHAPTER   XIII 
QUADRIC   SURFACES 

258.    The    Ellipsoid.      The    surface    represented    by    the 
equation 

is  called  an  ellipsoid.     Its  shape  is  best  investigated  by  tak- 
ing cross-sections  at  right  angles  to  the  axes  of  coordinates. 

Thus  the  coordinate  plane  Oyz  whose  equation  is  x  =  0  in- 
tersects the  ellipsoid  in  the  ellipse 

Any  other  plane  perpendicular  to  the  axis  Ox  (Fig.  125)  at 


FiQ.  125 

the  distance  h  <  a  from  the  plane  Oyz  intersects  the  ellipsoid 
in  an  ellipse  whose  equation  is 

i.e.  y^         ,         2'        _j 


'■('-5)"<'-S) 


242 


XIII,  §259]  QUADRIC  SURFACES  243 

Strictly  speaking  this  is  the  equation  of  the  cylinder  that  pro- 
jects the  cross-section  on  the  plane  Oyz.  But  it  can  also  be 
interpreted  as  the  equation  of  the  cross-section  itself,  referred 
to  the  point  (/i,  0,  0)  as  origin  and  axes  in  the  cross-section 
parallel  to  Oy  and  Oz. 

Notice  that  as  ^  <  a,  W^/o?,  and  hence  also  1  —  h^/a},  is  a  posi- 
tive proper  fraction.  The  semi-axes  6Vl  —  h^/o?,  c VI  —  h^/a} 
of  the  cross-section  are  therefore  less  than  h  and  c,  respec- 
tively. As  h  increases  from  0  to  a,  these  semi-axes  gradually 
diminish  from  h,  c  to  0. 

259.  Cross-Sections.  Cross-sections  on  the  opposite  side 
of  the  plane  Oyz  give  the  same  results;  the  ellipsoid  is  evi- 
dently symmetric  with  respect  to  the  plane  Oyz. 

By  the  same  method  we  find  that  cross-sections  perpendicu- 
lar to  the  axes  Oy  and  Oz  give  ellipses  with  semi-axes  dimin- 
ishing as  we  recede  from  the  origin.  The  surface  is  evidently 
symmetric  to  each  of  the  coordinate  planes.  It  follows  that 
the  origin  is  a  center,  i.e.  every  chord  through  that  point  is 
bisected  at  that  point.  In  other  words,  if  (x,  y,  z)  is  a  point 
of  the  surface,  so  is  (—a?,  —2/,  —z).  Indeed,  it  is  clear  from 
the  equation  that  if  (.t,  y,  z)  lies  on  the  ellipsoid,  so  do  the 
seven  other  points  {x,  y,  -z),  {x,  —y,  z),  (-x,  y,  z),  (x,  -y,  -z), 
(-  x,  y,  -  2),  (-  a;,  -  y,  z),  {-x,  -y,  -z).  A  chord  through 
the  center  is  called  a  diameter. 

It  follows  that  it  suffices  to  study  the  shape  of  the  portion  of 
the  surface  contained  in  one  octant,  say  that  contained  in  the  tri- 
hedral formed  by  the  positive  axes  Ox,  Oy,  Oz ;  the  remaining 
portions  are  then  obtained  by  reflection  in  the  coordinate  planes. 

The  ellipsoid  is  a  dosed  surface;  it  does  not  extend  to  in- 
finity ;  indeed  it  is  completely  contained  within  the  parallel- 
epiped with  center  at  the  origin  and  edges  2  a,  2  6,  2  c,  parallel 
to  Ox,  Oy,  Oz,  respectively. 


244 


SOLID  ANALYTIC  GEOMETRY-    [XIII,  §  260 


260.  Special  Cases.  In  general,  the  semi-axes  a,  b,  c  of  the 
ellipsoid,  i.e.  the  intercepts  made  by  it  on  the  axes  of  coordi- 
nates, are  different.  But  it  may  happen  that  two  of  them,  or 
even  all  three,  are  equal. 

In  the  latter  case,  i.e.  if  a  =  b  =  c,  the  ellipsoid  evidently 
reduces  to  a  sphere. 

If  two  of  the  axes  are  equal,  e.g.  if  6  =  c,  the  surface 

^  +  ^  +  51  =  1 
a"      b^      b^ 

is  called  an  ellipsoid  of  revolution  because  it  can  be  generated 
by  revolving  the  ellipse 


n      ' 


62 


1 


Fig.  126 


about  the   axis    Ox   (Fig.   126). 

Any  cross-section  at  right  angles 

to  Ox,  the  axis  of  revolution,  is  a 

circle,  while  the  cross-sections  at 

right  angles  to   Oy  and   Oz  are 

ellipses.     The  circular  cross-section  in  the  plane  Oyz  is  called 

the  equator ;  the  intersections  of  the  surface  with  the  axis  of 

revolution  are  the  poles. 

li  a  ^b  {a  being  the  intercept  on  the  axis  of  revolution), 
the  ellipsoid  of  revolution  is  called  prolate ;  if  a  <  b,  it  is 
called  oblate.  In  astronomy  the  ellipsoid  of  revolution  is 
often  called  spheroid,  the  surfaces  of  the  planets  which  are 
approximately  ellipsoids  of  revolution  being  nearly  spherical. 
Thus  for  the  surface  of  the  earth  the  major  semi-axis,  i.e.  the 
radius  of  the  equator,  is  3962.8  miles  while  the  minor  semi- 
axis,  i.e.  the  distance  from  the  center  to  the  north  or  south 
pole,  is  3949.6  miles. 


Xm,  §261]  QUADRIC  SURFACES  245 

261.  Surfaces  of  Revolution.  A  surface  that  cau  be  gen- 
erated by  the  revolution  of  a  plane  curve  about  a  line  in  the 
plane  of  the  curve  is  called  a  surface  of  revolution.  Any  such 
surface  is  fully  determined  by  the  generating  curve  and  the 
position  of  the  axis  of  revolution  with  respect  to  the  curve. 

Let  us  take  the  axis  of  revolution  as  axis  Ox,  and  let  the 
equation  of  the  generating  curve  be 

As  this  curve  revolves  about  Ox,  any 
point  P  of  the  curve  (Fig.  127)  de- 
scribes  a   circle   about    Ox  as   axis,  "et rTTisr , 

with  a  radius  equal  to  the  ordinate      /  f    i  / 

f(x)  of  the  generating   curve.     For  ^  \   '/ 

any  position  of  P  we  have  therefore  Fig.  127 

f +''■'= If  i^m 

and  this  is  the  equation  of  the  surface  of  revolution. 
Thus  if  the  ellipse 

revolves  about  the  axis  Ox,  we  find  since  y  =  ±  {h/a)^a^  —  o? 
for  the  ellipsoid  of  revolution  so  generated  the  equation 

2/2  +  22_5!(a2-a;2), 
a 

which  agrees  with  that  of  §  260. 

Any  section  of  a  surface  of  revolution  at  right  angles  to  the 
axis  of  revolution  is  of  course  a  circle ;  these  sections  are  called 
parallel  circles,  or  simply  parallels  (as  on  the  earth's  surface). 
Any  section  of  a  surface  of  revolution  by  a  plane  passing 
through  the  axis  of  revolution  is  called  a  meridian  section  ; 
it  consists  of  the  generating  curve  and  its  reflection  in  the  axis 
of  revolution. 


246  SOLID  ANALYTIC  GEOMETRY    [XIII,  §  261 

EXERCISES 

1.  An  ellipsoid  has  six/oci,  viz.  the  foci  of  the  three  ellipses  in  which 
the  ellipsoid  is  intersected  by  its  planes  of  symmetry.  Determine  the 
coordinates  of  these  foci :  (a)  for  an  ellipsoid  with  semi-axes  1,  2,  3 ; 
(6)  for  the  earth  (see  §  260)  ;  (c)  for  an  ellipsoid  of  semi-axes  10,  8,  1  ; 
(d)  for  an  ellipsoid  of  semi-axes  1,  1,  5. 

2.  Find  the  equations  of  the  surfaces  of  revolution  generated  by  re- 
volving the  following  curves  about  the  given  lines  : 

(a)  y  =  x",  about  the  axis  Ox. 

(b)  y-  =  X,  about  the  latus  rectum. 

(c)  a;2  _j_  y2  _  2  ac  =  0,  about  the  axis  Oy. 

(d)  x2  _  y2  _  1^  about  the  axis  Ox. 

3.  Find  the  equation  of  the  paraboloid  of  revolution  generated  by  the 
revolution  of  t-he  parabola  y^  =  4  ox  about  Ox. 

4.  Find  the  equation  of  a  torus,  or  anchor-ring,  i.e.  the  surface 
generated  by  the  revolution  of  a  circle  of  radius  a  about  a  line  in  its  plane 
at  the  distance  &  >  a  from  its  center. 

5.  Find  the  equation  of  the  surface  generated  by  the  revolution  of  a 
circle  of  radius  a  about  a  line  in  its  plane  at  the  distance  6  <  a  from  its 
center.  Is  the  appearance  of  this  surface  noticeably  different  from  the 
surface  of  Ex.  4  ?  What  happens  to  this  surface  when  6  =  0;  when  b  =  a? 

6.  Find  the  equation  of  the  surface  generated  by  the  revolution  of  the 
parabola  y^  =  iax  about :  (a)  the  tangent  at  the  vertex ;  (6)  the  latus 
rectum. 

7.  Find  the  equation  of  the  surface  generated  by  the  revolution  of  the 
hyperbola  xy  =  a^  about  an  asymptote. 

8.  Find  the  cone  generated  by  the  revolution  of  the  line  y  =  mx  -\-  b 
about:  (a)   Ox,  (6)   Oy. 

9.  How  are  the  following  surfaces  of  revolution  generated  ? 

(a)  y2+22=x*.         (6)  2x2-f-2y2-32=0.         (c)  x^-\-y^-z^-2x+i=0. 

10.  Find  the  equation  of  the  surface  generated  by  the  revolution  of 
the  ellipse  x^  4-  4  y2  _  4  a;  =  o  :  (a)  about  the  major  axis  ;  (b)  about  the 
minor  axis ;  (c)  about  the  tangent  at  the  origin. 


XIII,  §  263] 


QUADRIC  SURFACES 


247 


262.   Hyperboloid  of  One  Sheet.     The  surface  represented 
by  the  equation 


is  called  a  hyperboloid  of  one  sheet  (Fig.  128).    The  intercepts 


Fig.  128 

on  the  axes  Ox,  Oy  are  ±  a,  ±  6 ;  the  axis  Oz  does  not  intersect 
the  surface. 

263.   Cross-Sections.     The  plane  Oxy  intersects  the  surface 
in  the  ellipse 

cross-sections   perpendicular  to   Oz    give  ellipses   with  ever- 
increasing  semi-axes. 

The  planes  Oyz  and  Ozx  intersect  the  surface  in  the  hyperbolas 


^_?.  —  1     ^__  —  1 
62     ^2~    '    a2      c2~ 


Any  plane  perpendicular  to  Ox,  at  the  distance  h  from  the 
origin,  intersects  the  hyperboloid  in  a  hyperbola,  viz. 


f 


"(•-i)  <'-3 


1. 


248  SOLID  ANALYTIC  GEOMETRY  .  [XIII,  §  263 

As  long  as  ^  <  a  this  hyperbola  has  its  transverse  axis  parallel 
to  Oy  while  for  1i'>a  the  transverse  axis  is  parallel  to  Oz ;  for 
h  =  a  the  equation  reduces  to  y''-/h'^  —  z^l&  =  0  and  represents 
t"wo  straight  lines,  viz.  the  parallels  through  (a,  0,  0)  to  the 
asymptotes  of  the  hyperbola  yV^*  ~  ^V^'^  =  ^  which  is  the 
intersection  of  the  surface  with  the  plane  Oyz. 

Similar  considerations  apply  to  the  cross-sections  perpen- 
dicular to  Oy. 

The  hyperboloid  has  the  same  properties  of  symmetry  as  the 
ellipsoid  (§  259)  ;  the  origin  is  a  center ,  and  it  suffices  to  inves- 
tigate the  shape  of  the  surface  in  one  octant. 

264.  Hyperboloid  of  Revolution  of  One  Sheet.    If  in  the 

hyperboloid  of  one  sheet  we  have  a  =  b,  the  cross-sections  per- 
pendicular to  the  axis  Oz  are  all  circles  so  that  the  surface  can 
be  generated  by  the  revolution  of  the  hyperbola 

about  Oz.  Such  a  surface  is  called  a  hyperboloid  of  revolution 
of  one  sheet. 

265.  Other  Forms.     The  equations 

a*     6»     c2       '        a*     b^     c^ 
also  represent  hyperboloids  of  one  sheet  which  can  be  investi- 
gated as  in  §§  262-264.     In  the  former  of  these  the  axis  Oy,  in 
the  latter  the  axis  Ox,  does  not  meet  the  surface. 
Every  hyperboloid  of  one  sheet  extends  to  infinity. 

266.  Hyperboloid  of  Two  Sheets.  The  surface  represented 
by  the  equation 

a"     b^     c2 
is  called  a  hyperboloid  of  two  sheets  (Fig.  129). 


XIII,  §  269] 


QUADRIC  SURFACES 


249 


The  intercepts  on  Ox  are  ±  a ;  the  axes  Oy,  Oz  do  not  meet 
the  surface. 

267.   Cross-Sections.     The  cross-sections  at  right  angles  to 
Ox,  at  the  distance  h  from  the  origin  are 


'■('-i)"<'-S)" 


these  are  imaginary  as  long  as  h  <  a; 
for  h>a  they  are  ellipses  with  ever- 
increasing  semi-axes  as  we  recede  from 
the  origin. 

The  cross-sections  at  right  angles  to  Oy 
and  Oz  are  hyperbolas. 

The  hyperboloid  of  two  sheets,  like  that  of  one  sheet  and 
like  the  ellipsoid,  has  three  mutually  rectangular  planes  of 
symmetry  whose  intersection  is  therefore  a  center. 

The  surfaces 

z^      -,  x^ 


Fig.  129 


^2  ^      Z^_  ^ 


^=1 


are  hyperboloids  of  two  sheets,  the  former  being  met  by  Oy, 
the  latter  by  Oz,  in  real  points. 

The  hyperboloid  of  two  sheets  extends  to  infinity. 

268.  Hyperboloid  of  Revolution  of  Two  Sheets.    If  6  =  c 

in  the  equation  of  §  266,  the  cross-sections  at  right  angles  to  Ox 
are  circles  and  the  surface  becomes  a  hyperboloid  of  revolution 
of  two  sheets. 

269.  Imaginary  Ellipsoid.     The  equation 


=  1 


a2      b^     c2 

is  not  satisfied  by  any  point  with  real  coordinates.    It  is  some- 
times said  to  represent  an  imaginary  ellipsoid. 


250 


SOLID  ANALYTIC  GEOMETRY     [XIII,  §270 


270.  The  Paraboloids. 


—  4-  ^  =  2  cz. 


'2^62 


-^  =  2c2, 


The  surfaces 

a-'     0==  a2 

which  are  called  the  elliptic  paraboloid  (Fig.  130)  and  hyper- 
bolic paraboloid  (Fig.  131),  respectively,  have  each  only  two 
planes  of  symmetry,  viz.  the  planes  Oyz  and  Ozx.  We  here 
assume  that  c=^0.     The  cross-sections  at  right  angles  to  the 


Fig.  130 


axis  Oz  are  evidently  ellipses  in  the  case  of  the  elliptic  parab- 
oloid, and  hyperbolas  in  the  case  of  the  hyperbolic  paraboloid. 
The  plane  Oxy  itself  has  only  the  origin  in  common  with  the 
elliptic  paraboloid ;  it  intersects  the  hyperbolic  paraboloid  in 
the  two  lines  x^/a?'  —  y^jW-  —  0,  i.e.  y  =  ±  hx/a. 

The  intersections  of  the  elliptic  paraboloid  (Fig.  130)  with 
the  planes  Oyz  and  Ozx  are  parabolas  with  Oz  as  axis  and  0  as 
vertex,  opening  in  the  sense  of  positive  2;  if  c  is  positive,  in  the 
sense  of  negative  z  if  c  is  negative.  Planes  parallel  to  these 
coordinate  planes  intersect  the  elliptic  paraboloid  in  parabolas 
with  axes  parallel  to  Oz,  but  with  vertices  not  on  the  axes  Oa;, 
Ot/,  respectively. 

For  the  hyperbolic  paraboloid  (Fig.  131),  which  is  saddle- 
shaped  at  the  origin,  the  intersections  with  the  planes  Oyz  and 


XIII,  §  273] 


QUADRIC  SURFACES 


251 


Ozx  are  also  parabolas  with  Oz  as  axis ;  if  c  is  positive  the 
parabola  in  the  plane  Oyz  opens  in  the  sense  of  negative  z,  that 
in  the  plane  Ozx  opens  in  the  sense  of  positives.  Similarly 
for  the  parallel  sections. 

271.  Paraboloid  of  Revolution.     If  in  the  equation  of  the 
elliptic  paraboloid  we  have  a  =  6,  it  reduces  to  the  form 

x^-\-y^  =  2pz. 

This  represents  a  surface  of  revolution,  called  the  paraboloid  of 
revolution.  This  surface  can  be  regarded  as  generated  by  the 
revolution  of  the  parabola  y^  =  2pz  about  the  axis  Oz. 

272.  Elliptic  Cone.    The  surface  represented  by  the  equation 


^     f 
a^     b^ 


=  0 


is  an  elliptic  cone,  with  the  origin  as  vertex  and  the  axis  Oz  as 
axis  (Fig.  132). 

The  plane  Oxy  has  only  the  origin  in 
common  with  the  surface.  Every  parallel 
plane  z  =  k,  whether  Ic  be  positive  or  negative, 
intersects  the  surface  in  an  ellipse,  with 
semi-axes  increasing  proportionally  to  k. 

The  plane  Oyz,  as  well  as  the  plane  Ozx, 
intersects  the  surface  in  two  straight  lines 
through  the  origin.  Every  plane  parallel  to 
Oyx  or  to  Ozx  intersects  the  surface  in  a 
hyperbola.  Fia.  132 

273.  Circular  Cone.  If  in  the  equation  of  the  elliptic  cone 
we  have  a  =  b,  the  cross-sections  at  right  angles  to  the  axis  Oz 
become  circles.     The  cone  is  then  an  ordinary  circular  cone,  or 


252  SOLID  ANALYTIC  GEOMETRY     [XIII,  §  273 

cone  of  revolution,  which  can  be  generated  by  the  revolution 
of  the  line  y  =  (a/c)z  about  the  axis  Oz.  Putting  a/c  =  m  we 
can  write  the  equation  of  a  cone  of  revolution  about  Oz,  with 
vertex  at  0,  in  the  form 

274.  Quadric  Surfaces.  The  ellipsoid,  the  two  hyper- 
boloids,  the  two  paraboloids,  and  the  elliptic  cone  are  called 
quadric  surfaces  because  their  cartesian  equations  are  all  of 
the  second  degree. 

Let  us  now  try  to  determine,  conversely,  all  the  various  loci 
that  can  be  represented  by  the  general  equation  of  the  second 
degree 

Ax^  +  By^  +Cz'^  +  2  Dyz  -h  2  Ezx  -h  2  Fxy 

'Jt2Gx-\-2Hy  +  2Iz  +  J=0. 

In  studying  the  equation  of  the  second  degree  in  x  and  y 
(§  253)  it  was  shown  that  the  term  in  xy  can  always  be 
removed  by  turning  the  axes  about  the  origin  through  a  cer- 
tain angle.  Similarly,  it  can  be  shown  in  the  case  of  three 
variables  that  by  a  properly  selected  rotation  of  the  coordinate 
trihedral  about  the  origin  the  terms  in  yx,  zx,  xy  can  in  general 
all  be  removed  so  that  the  equation  reduces  to  the  form 

(1)         Ax^  +  By^  +  C^^  +  2  Gx  +  2  JJy  +2  J»  -H  «/■=  0. 

This  transformation  being  somewhat  long  will  not  be  given 
here.  We  shall  proceed  to  classify  the  surfaces  represented 
by  equations  of  the  form  (1). 

275.  Classification.  The  equation  (1)  can  be  further  sim- 
plified by  completing  the  squares.  TJiree  cases  may  be  distin- 
guished according  as  the  coefficients  A,  By  C  are  all  three  differ- 
ent from  zero,  one  only  is  zero,  or  two  are  zero. 


XIII,  §275]  QUADRIC  SURFACES  253 

Case  (a):  A  =^  0,  B  ::^0,  C^  0.     Completing  the  squares  in 
X,  y,  z  we  find 

Referred  to  parallel  axes  through  the  point  (—  G/A,  —  H/B, 
—  I/C)  this  equation  becomes 

(2)  Ax''-\-By^-\-Cz^  =  J,. 

Case  (6)  :  A=^0,  B=^0,  0=0.     Completing  the  squares  in  x 
and  y  we  find 


(-i) 


^^^  B  A       B  ' 


If  1=^0,  we  can  transform  to  parallel  axes  through  the  point 
(—G/A,  —  H/B,  J2/2  I)  so  that  the  equation  becomes 

(3)  Ax"  +  By''-i-2Iz  =  0. 

If,  however,  7=0,  we  obtain  by  transforming  to  the  point 
(-G/A,-II/B,0) 

(3')  Ax'-\-By^=J,. 

Case  (c)  :  A^O,  B  =  0,  C  =  0.     Completing  the  square  in 
X  we  have 

^2  6?2 


-(-SJ 


+  2  Hy  +  2  Iz  =  ^  -J=J^. 


If  H  and  /  are  not  both  zero,  we  can  transform  to  parallel 
axes  through  the  point  (—  G/A,  J^/2  H,  0)  or  through  (—  G/A, 
0,  J3/2  /)  and  find 

(4)  .  Ax'  +  2Hy  +  2Iz  =  0. 

If  Zr=  0  and  7=  0,  we  transform  to  the  point  (—  G/A,  0,  0) 
so  that  we  find 
(4')  Ax''=J,. 


254  SOLID  ANALYTIC  GEOMETRY    [XIII,  §  276 

276.  Squared  Terms  all  Present,  Case  (a).  We  proceed  to 
discuss  the  loci  represented  by  (2).  If  J^  ^  0,  we  can  divide 
(2)  by  t/i  and  obtain  : 

(a)  if  A/Ji ,  B/J^ ,  C/Ji  are  positive,  an  ellipsoid  (§  258) ; 

(fi)  if  two  of  these  coefficients  are  positive  while  the  third 
is  negative,  a  Jiyperholoid  of  one  sheet  (§  262) ; 

(y)  if  one  coefficient  is  positive  while  two  are  negative,  a 
hyperboloid  of  tivo  sheets  (§  266) ; 

(8)  if  all  three  coefficients  are  negative,  the  equation  is  not 
satisfied  by  any  real  point  (§  269) ;    . 

If  J^  =  0,  the  equation  (2)  represents  an  elliptic  cone  (§  272) 
unless  A,  B,  C  all  have  the  same  sign,  in  which  case  the  origin 
is  the  only  point  represented. 

277.  Case  (b).  The  equation  (3)  of  §275  evidently  fur- 
nishes the  two  paraboloids  (§  270) ;  the  paraboloid  is  elliptic  if 
A  and  B  have  the  same  sign;  it  is  hyperbolic  if  A  and  B  are  of 
opposite  sign. 

The  equation  (S*),  since  it  does  not  contain  z  and  hence  leaves 
z  arbitrary,  represents  the  cylinder,  with  generators  parallel  to  Oz, 
passing  through  the  conic  Ax^  +  By"^  =^  J^,  As  ^  and  B  are 
assumed  different  from  zero,  this  conic  is  an  ellipse  if  AfJ^  and 
5/J2  are  both  positive,  a  hyperbola  if  AjJ^  and  ^/c/,  are  of 
opposite  sign,  and  it  is  imaginary  if  AjJ^  and  -B/Jg  are  both 
negative.  This  assumes  J^  4^  0.  If  J^  —  0,  the  conic  degen- 
erates into  two  straight  lines,  real  or  imaginary ;  the  cylinder 
degenerates  into  two  planes  if  the  lines  are  real. 

278.  Case  (c).  There  remain  equations  (4)  and  (4').  To  sim- 
plify (4)  we  may  turn  the  coordinate  trihedral  about  Ox  through 
an  angle  whose  tangent  is  —  HII\  this  is  done  by  putting 

-^H'+P  ^/H'  +  P 


XIII,  §  278]  QUADRIC  SURFACES  255 

our  equation  then  becomes 


It  evidently  represents  a  parabolic  cylinder,  with  generators 
parallel  to  Oy. 

Finally,  the  equation  (4')  is  readily  seen  to  represent  two 
planes  perpendicular  to  Ox,  real  or  imaginary,  unless  J3  =  0, 
in  which  case  it  represents  the  plane  Oyz. 

EXERCISES 

1.  Name  and  locate  the  following  surfaces  : 

(a)  a;2  +  2  ?/2  +  3^2  =  4.  (Jb)  x"^  +  y'^  -  hz -Q  =  0. 

(c)   x^ -  y'^  +  z"^  =  4..  (d)  x2-y^  +  z^-\-Sz  +  6  =  0. 

(e)   2?/2 -4:^2  _  5=0.  (/)  2a;2  +  y2_|.3^2  +  5_0. 

(g)  6;s2  +  2x2  =  10.  ih)  z^-9  =  0. 

(i)    x2-y  +  l  =  0.  0*)  x^-y^-z^  +  6z  =  9. 

(k)  x^  +  Sy'^  +  z"^  -j-  4  z  -\-  4  =  0.         {I)  z'^-hy -9  =  0. 

2.  The  cone 

x2/a2  +  yyb^  -  02/c2  =  0 

is  called  the  asymptotic  cone  of  the  hyperboloid  of  one  sheet 

a;2/a2  +  ^2/52  _  ^2/c2  =  1. 
Show  that  as  z  increases  the  two  surfaces  approach  each  other,  i.e.  they 
bear  a  relation  similar  to  a  hyperbola  and  its  asymptotes. 

3.  What  is  the  asymptotic  cone  of  the  hyperboloid  of  two  sheets  ? 

4.  Show  that  the  intersection  of  a  hyperboloid  of  two  sheets  with  any 
plane  actually  cutting  the  surface  is  an  ellipse,  parabola,  or  hyperbola. 
Determine  the  position  of  the  plane  for  each  conic. 

5.  Show  that  in  general  nine  points  deterinine  a  quadric  surface  and 
that  the  equation  may  be  written  as  a  determinant  of  the  tenth  order 
equated  to  zero. 

6.  Show  that  the  surface  inverse  to  the  cylinder  x^  -\-  y^  =  a^,  with 
respect  to  the  sphere  ^2  4-  y2  +  ^2  _  ^52^  ig  the  torus  generated  by  the  rev- 
olution of  the  circle  (y  —  a/2)2  +  z^  =  a^  about  the  axis  Ox. 

7.  Determine  the  nature  of  the  surface  xyz  =  a^  by  means  of  cross- 
sections. 


256  SOLID  ANALYTIC  GEOMETRY      [XIII,  §  279 

279.   Tangent  Plane  to  the  Ellipsoid.     The  plane  tangent 
to  the  ellipsoid 

a"     h^     c^ 

can  be  found  as  follows  (compare  §§  255,  256).  The  equa- 
tions of  the  line  joining  any  two  given  points  {x^  y^  z^  and 
{^ ,  2/2 ,  22)  are 

x=^x^-{-k{Xi  —  x^\   2/  =  2/i  +  %2-yi),   z  =  2!i  +  A:(22-2;i). 

This  line  will  be  tangent  to  the  ellipsoid  if  the  quadratic 
in  k 

[_x,  +  k{x^-x,)J      [yi-^k(y,-y,)J      [z,  +  k(z,-z,)y  ^^ 
a«  "^  6«  "^  c« 

has  equal  roots.     Writing  this  quadratic  in  the  form 

nx^-x,y  _^  (y,-y,y  _^  (z,-z,y\t 

\_      0}  b^  c*      J 

o^xiix^-x^)     yijy^-y,)  ■  z^jz^-z,)!.     fx,^    y,^    z,^    ^\     ^ 
\_       a^  6*  o"       \   ^^a^^h^^ c"       J       ' 

we  find  the  condition 

a2  "^  62  "^  c2        ;     \o?       62       c2        )\ 

J{x,-x,y     (y,-y,y  ,  fe-^OnW  I  yx'  ,  z,^     .\ 
L      a2  6*  c2       JVa^      62      c2        y 

If  now  we  keep  the  point  {x^ ,  2/1 ,  ^i)  fixed,  but  let  the  point 
(^2)  .V2)  2:2)  vary  subject  to  this  condition,  it  will  describe  the 
cone,  with  vertex  (oq ,  2/1 ,  2i),  tangent  to  the  ellipsoid ;  to  indi- 
cate this  we  shall  drop  the  subscripts  of  x^,  y^y  z^.  If,  in 
particular,  the  point  {x^ ,  y^ ,  Zi)  be  chosen  on  the  ellipsoid,  we 
have 

^'  +  .^'  +  !l  =  i, 
a2      62      c2 


XIII,  §281]  QUADRIC  SURFACES  257 

and  the  cone  becomes  the  tangent  plane.     The  equation  of  the 
tangent  plane  to  the  ellipsoid  at  the  point  {x^^ ,  y^ ,  z^  is,  therefore : 

a2       &2  "1"  c2 

280.  Tangent  Planes  to  Hj^erboloids.  In  the  same  way 
it  can  be  shown  that  the  tangent  planes  to  the  hyperboloids 

a2      62     c2        '    a2     52      (.2 
at  (a;i,2/i,2i)  are 

a2  62  c2  >        ^2  52  ^2 

By  an  equally  elementary,  but  somewhat  longer,  calculation 
it  can  be  shown  that  the  tangent  plane  to  the  quadric  surface 

Ax''  +  By^  +  C0'  +  2  D?jz  +  2Ezx  +  2  Fxy 

+  2Gx-\-2Hy  +  2Iz-\-J=0 
at  (.Tj ,  2/1 J  2;i)  is : 

AxyX  4-  %i2/  +  C2i2  +  Z)(yi2;  +  ^iV)  +  ^(^^i.^  +  a?i2;)  +  F{x,y  +  ^/lO;) 
+  6?(x,  +  ir)  +  ^(2/1  +  2/)  + /(^i  +  ^)  +  J^=  0, 

In  particular,  the  tangent  planes  to  the  paraboloids 

tj^yl^2cz,    t-t=.2cz 
a2      62  a"      62 

are 

281.  Ruled  Surfaces.  A  surface  that  can  be  generated  by 
the  motion  of  a  straight  line  is  called  a  ruled  surface;  the  line 
is  called  the  generator. 

The  plane  is  a  ruled  surface.  Among  the  quadric  surfaces 
not  only  the  cylinders  and  cones  but  also  the  hyperboloid  of 
one  sheet  and  the  hyperbolic  paraboloid  are  ruled  surfaces. 


258 


SOLID  ANALYTIC  GEOMETRY      [XIII,  §  282 


282.  Rulings  on  a  Hyperboloid  of  One  Sheet.    To  show 
this  for  the  hyperboloid 

a"     b^     c2       ' 
we  write  the  equation  in  the  form 

and  factor  both  members : 


Ci-t){l-H^!X'-l} 


It  is  then  apparent  that  any  point  whose  coordinates  satisfy 
the  two  equations 

be        \       aj     b     a     k\       aj 

where  k  is  an  arbitrary  parameter,  lies 
on  the  hyperboloid.  These  two  equa- 
tions represent  for  every  value  of  A;  (:^  0) 
a  straight  line.  The  hyperboloid  of  one 
sheet  contains  therefore  the  family  of 
lines  represented  by  the  last  two  equa- 
tions with  variable  A:. 

In  exactly  the  same  way  it  is  shown  that  the  same  hyper- 
boloid also  contains  the  family  of  lines 

be        V        aJ     b     c     k'\       aJ 


Fig.  133 


Thus  every  hyperboloid  of  one  sheet  contains  two  sets  of  recti- 
linear generators  (Fig.  133). 


XIII,  §  283] 


QUADRIC  SURFACES 


259 


283.  Rulings  on  a  Hyperbolic  Paraboloid.    The  hyperbolic 
paraboloid  (Fig.  134) 


x^      y^ 


=  2C2 


also  contains  two  sets  of  recti- 
linear  generators^  namely, 

a      b  a     b     k 

and 

a     b  a     b     k' 


Fig.  134 


EXERCISES 

1.  Derive  the  equation  of  the  tangent  plane  to  : 

(a)  the  elliptic  paraboloid  ;   (b)  the  hyperbolic  paraboloid  ; 
(c)  the  elliptic  cone. 

2.  The  line  perpendicular  to  a  tangent  plane  at  a  point  of  contact  is 
called  the  normal  line.  Write  the  equations  of  the  tangent  planes  and 
normal  lines  to  the  following  quadric  surfaces  at  the  points  indicated : 

(a)  a;V9  +  y^i  -  ^716  =  1,  at  (3,  -  1,  2) ; 
(6)  a;2  + 2  2/2 +  2^2^10,  at  (2,1,  -2); 
(c)  a;2  +  2  y2  _  2  ;22  ^  0,  at  (4,  1,  3) ;  (d)  x^-Sy^-z  =  0,  at  the  origin. 

3.  Show  that  the  cylinder  whose  axis  has  the  direction  cosines  I,  rn,  n 
and  which  is  tangent  to  the  ellipsoid  x^/a^  +  y^/b^  +  z^/c^  =  1,  is 

w    b^    cy     U'"   &2    c2;U'   &■'  c2    /    • 

4.  Show  that  the  plane  Ix  -^  my  +  nz  =  Vl'-^a^  +  m^b'^  +  n^c^  is  tangent 
to  the  ellipsoid  x^/a"^  +  y'^/b'^  +  z^/c"^  =  1. 

5.  Show  that  the  locus  of  the  intersection  of  three  mutually  perpen- 
dicular tangent  planes  to  the  ellipsoid  a:2/a2  -f  2/2/52  _f.  ^2/02  =  1,  is  the 
sphere  (called  director  sphere)  x^  -{■  y"^  +z^  =  a^  +  b^  +  c2. 


260  SOLID  ANALYTIC  GEOMETRY     [XIII,  §  283 

6.  Show  that  the  elliptic  cone  is  a  ruled  surface. 

7.  Show  that  any  two  linear  equations  which  contain  a  parameter 
represent  the  generating  line  of  a  ruled  surface.  What  surfaces  are  gen- 
erated by  the  following  lines  ? 

(a)  x-y-\-kz  =  0,x-\-y-z/k  =  Q;  (6)  3  a;  -  4  y  =  A;,  (3  a;+4  y)k=l ; 
(c)  X  -  y  +  3  A-2  =  3  k,  k{x  +  y)—  2  =  3. 

8.  Show  that  every  generating  line  of  the  hyperbolic  paraboloid 
a;2/a2  _  y'ljifi  =  2  C2  is  parallel  to  one  of  the  planes  x^/ct^  —  y'^/h'^  =  0. 

284.  Surfaces  in  General  When  it  is  required  to  deter- 
mine the  shape  of  a  surface  from  its  cartesian  equation 

the  most  effective  methods,  apart  from  the  calculus,  are  the 
transformation  of  coordinates  and  the  taking  of  cross-sections, 
generally  (though  not  necessarily  always)  at  right  angles  to 
the  axes  of  coordinates.  Both  these  methods  have  been  ap- 
plied repeatedly  to  the  quadric  surfaces  in  the  preceding 
articles. 

285.  Cross-Sections.  The  method  of  cross-sections  is  ex- 
tensively used  in  the  applications.  The  railroad  engineer  de- 
termines thus  the  shape  of  a  railroad  dam ;  the  naval  architect 
uses  it  in  laying  out  his  ship ;  even  the  biologist  uses  it  in  con- 
structing enlarged  models  of  small  organs  of  plants  or  animals. 

286.  Parallel  Planes.  When  the  given  equation  contains 
only  one  of  the  variables  x,  y,  z,  it  represents  of  course  a  set  of 
parallel  planes  (real  or  imaginary),  at  right  angles  to  one  of 
the  axes.     Thus  any  equation  of  the  form 

F(x)=0 

represents  planes  at  right  angles  to  Ox,  of  which  as  many  are 
real  as  the  equation  has  real  roots. 


XIII,  §  290]  QUADRIC   SURFACES  261 

287.  Cylinders.  When  the  given  equation  contains  only  two 
variables  it  represents  a  cylinder  at  right  angles  to  one  of  the 
coordinate  planes.     Thus  any  equation  of  the  form 

F(x,y)=0 
represents  a  cylinder  passing  through  the  curve  F{x,  y)  =  0  in 
the  plane  Oxy,  with  generators  parallel  to  Oz.    If,  in  particular, 
F(x,  y)  is  homogeneous  in  x  and  y,  i.e.  if  all  terms  are  of  the 
same  degree,  the  cylinder  breaks  up  into  planes. 

288.  Cones.  When  the  given  equation  F(x,  y^  z)=0  is 
homogeneous  in  x,  y,  and  z,  i.e.  if  all  terms  are  of  the  same 
degree,  the  equation  represents  a  general  cone,  with  vertex  at 
the  origin.  For  in  this  case,  if  (x,  y,  z)  is  a  point  of  the  sur- 
face, so  is  the  point  (kx,  ky,  kz),  where  k  is  any  constant;  in 
other  words,  if  P  is  a  point  of  the  surface,  then  every  point  of 
the  line  OP  belongs  to  the  surface ;  the  surface  can  therefore 
be  generated  by  the  motion  of  a  line  passing  through  the  origin 

289.  Functions  of  Two  Variables.  Just  as  plane  curves  are 
used  to  represent  functions  of  a  single  variable,  so  surfaces  can 
be  used  to  represent  functions  of  two  variables.  Thus  to  obtain 
an  intuitive  picture  of  a  given  function  f(x,  y)  we  may  con- 
struct a  model  of  the  surface 

such  as  the  relief  map  of  a  mountainous  country.     The  ordi- 
nate z  of  the  surface  represents  the  function. 

290.  Contour  Lines.  To  obtain  some  idea  of  such  a  surface 
by  means  of  a  plane  drawing  the  method  of  contour  lines  or 
level  lines  can  be  used.  This  is  done,  e.g.,  in  topographical 
maps.  The  method  consists  in  taking  horizontal  cross-sections 
at  equal  intervals  and  projecting  these  cross-sections  on  the  hori- 
zontal plane.  Where  the  level  lines  crowd  together  the  surface 
is  steep ;  where  they  are  relatively  far  apart  the  surface  is  flat. 


262 


SOLID  ANALYTIC  GEOMETRY      [XIII,  §  290 


EXERCISES 
1.   What  surfaces  are  represented  by  the  following  equations  ? 


(a)  Ax  +  By-{-  C  =  0. 

(c)  y^  +  z'^^a^. 

(e)  zx-a^, 

(g)  x«- 3x2-3;+ 3  =  0. 

(0   y  =  x2-x-6. 

{k)  x2  +  2  1/2  =  0. 

(m)  x2  -  J/2  =  2-2. 

(0)   (x-l)(y-2)(;^-3)  =  0. 


(6)  xcos/3  +  ysin  j3=p. 

(/);?2  =  4ay. 

(h)  xyz  =  0. 

U)  yz^-9y  =  0. 

(0   x^  =  yz. 

(n)  y^-{-2z^-\-izx  =  0. 


2.  Determine  the  nature  of  the  following  surfaces  by  sketching  the 
contour  lines : 

(a)  z=x  +  y.  (6)   z  =  xy.  (c)  z  =  y/x.  (d)  z  =x2  +  y, 

(e)  2=x2-y2+4.     (/)  2  =  x2.  (g)    z  =  x^+y^-ix.    (h)  z  =  xy~x. 

(i)  z  =  2'.  (j)    y=«2_4x.    (^-)  y  =  3  ^2  +  x2.        (Z)    «=3  x+y2. 

3.  The  Cassinian  ovals  (§  178)  are  contour  lines  of  what  surface  ? 

4.  What  can  be  said  about  the  nature  of  the  contour  lines  of  a  sur- 
face z=f(x)  ?  Discuss  in  particular  :  (a)  2  =  x2  —  9  ;  (b)  z  =  x^  —  8; 
(c)  y  =  z^-{-2z. 

291.  Rotation  of  Coordinate  Trihedral.  To  transform  the 
equation  of  a  surface  from  one  coordinate  trihedral  Oxyz  to  another 
Ox'y'z',  with  the  same  origin  O,  we 
must  find  expressions  for  the  old  co- 
ordinates X,  y,  z  of  any  point  P  in  terms 
of  the  new  coordinates  x',  y',  z'.  We 
here  confine  ourselves  to  the  case  when 
each  trihedral  is  trirectangular ;  this  is 
the  case  of  orthogonal  transformation^ 
or  orthogonal  substitution. 

Let  ^1 ,  mi ,  ni  be  the  direction  cosines 
of  the  new  axis  Ox'  with  respect  to  the 
old  axes  Ox,  Oy,  Oz  (Fig.  136) ;  similarly 
h,  m2 ,  W2  those  of  Oy',  and  ^3 ,  m^ ,  na  those  of  Oz'.    This  is  indicated  by 
the  scheme 


Fig.  135 


XIII,  §293]  QUADRIC  SURFACES  263 


X' 


h^  +  mi2  +  ni2  =  1, 

h^  +  mi^  +  7122  =  1, 

Zs^  +  m32  +  W32  =  1, 

Zi^  +  22^  +  Za^  =  1, 

mi2  +  m22  +  m32  =  ] 

Zl      Z2      Z3 

Wll      Wi2      WI3 

wi     n2     ns 

which  shows  at  the  same  time  that  then  the  direction  cosines  of  the  old 
axis  Ox  with  respect  to  the  new  axes  Ox',  Oy' ,  Oz'  are  Zi ,  ^2 ,  Z3  ,  etc. 

292.  The  nine  direction  cosines  Zi ,  Z2 ,  •••  W3  are  sufficient  to  determine 
the  position  of  the  new  trihedral  Ox'y'z'  with  respect  to  the  old.  But 
these  nine  quantities  cannot  be  selected  arbitrarily  ;  they  are  connected  by 
six  independent  relations  which  can  be  written  in  either  of  the  equivalent 
forms 

Z2Z3  +  m2m3  +  712^3  =  0, ' 

(1)  Z22  +  m22  +  7122  =  1,  Z3Z1  +  mzlUi  +  WsWi  =  0, 

Z1Z2  +  WliW2  4-  WiW2  =  0, 

wiini  +  miUi  +  WI3W3  =  0, 
(1')        mi2  +  m22  +  m32  =  1,  riih  +  n2Z2  +  W3Z3  =  0, 

Wi2  +  W22  4-  W32  =  1 ,  limi  +  Z2TO2  +  hmz  =  0. 

The  meaning  of  these  equations  follows  from  §§  202  and  205.  Thus 
the  first  of  the  equations  (1)  expresses  the  fact  that  Zi,  mi,  wi  are  the 
direction  cosines  of  a  line,  viz.  Ox'  ;  the  last  of  the  equations  (1')  ex- 
presses the  perpendicularity  of  the  axes  Ox  and  Oy  ;  and  so  on. 

293.  If  X,  y,  z  are  the  old,  x',  y\  z'  the  new  coordinates  of  one  and 
the  same  point,  we  find  by  observing  that  the  projection  on  Ox  of  the 
radius  vector  of  P  is  equal  to  the  sum  of  the  projections  on  Ox  of  its 
components  x',  j/',  z'  (§  199),  and  similarly  for  the  projections  on  Oy 
and  Oz : 

X  =  Zix'  +  hyi  +  hz', 

(2)  y  =  Ttiix'  +  miy'  +  mzz', 
z  =  mx'  +  n2y'  +  nzz'. 

Indeed,  these  relations  can  be  directly  read  off  from  the  scheme  of 
direction  cosines  in  §  291. 

Likewise,  projecting  on  Ox',  Oy',  Oz',  we  find 

x'  =  Zix  +  m\y  +  n\z, 
(2')  y'  =  hx  +  nny  +  n2Z, 

z'  =  I3X  +  mzy  +  nzz. 


264  SOLID  ANALYTIC  GEOMETRY     [XIII,  §  293 

As  the  equations  (2),  by  means  of  which  we  can  transform  the  equation 
of  any  surface  from  one  rectangular  system  of  coordinates  to  any  other 
with  the  same  origin,  give  x,  y,  z  as  linear  functions  of  x\  y\  «',  it  follows 
that  8uch  a  transformation  cannot  change  the  degree  of  the  equation  of 
the  surface. 

294.  The  equation  (2')  must  of  course  result  also  by  solving  the  equa- 
tions (2)  for  x',  2/',  z\  and  vice  versa.     Putting 

l\      h      h 

wii     »n2     Wl3    =  A 

ni      n2      W3 

solving  (2)  for  x',  j/',  «',  and  comparing  the  coefficients  of  x,  y,  z  with 
those  in  (2')  we  find  the  following  relations  : 

Dll  =  7712713  —  7713712  ,         DMi  =  7l2?3  —  W3Z2  »       DtIi  =  litn^  —  Z3WI2  1      ©tC. 

Squaring  and  adding  the  first  three  equations  and  applying  the  re- 
lations (1)  we  find  :  Z)2  =  1. 

By  §  226,  D  can  be  interpreted  as  six  times  the  volume  of  the  tetrahe- 
dron whose  vertices  are  the  origin  and  the  points  x',  y',  z'  in  Fig.  135,  i.e. 
the  intei-sections  of  the  new  axes  with  the  unit  sphere  about  the  origin. 
The  determinant  gives  this  volume  with  the  sign  +  or  —  according  as  the 
trihedral  Ox'y'z'  is  superposable  or  not  (in  direction  and  sense)  to  the 
trihedral  Oxyz  (see  §  295).     It  follows  that  D=±l  and 

li=±  (m27i3  —  W3W2),     wii  =  i  (712I3  -  713/2),     ni=±  {hmz  —  hmi)^ 
h  —  ±  (w»37ii  —  ?7»in3),     «i2  =  ±  {nzh  —  nih),     7i2  =  ±  Chrni  —  hmz)^ 
^3  =±  (77ii7i2  — wi27ii),     ^3  =  ±  (7ii?2  —  Wa/i),     n^  =  ±  (hm^  —  hmi) , 
the  upper  or  lower  signs  to  be  used  according  as  the  trihedrals  are  super- 
posable or  not. 

295.  A  rectangular  trihedral  Oxyz  is  called  right-handed  if  the  rotation 
that  turns  Oy  through  90°  into  Oz  appears  counterclockwise  as  seen  from 
Ox  ;  otherwise  it  is  called  left-handed.  In  the  present  work  right-handed 
sets  of  axes  have  been  used  throughout. 

Two  right-handed  as  well  as  two  left-handed  rectangular  trihedrals  are 
superposable ;  a  right-handed  and  a  left-handed  trihedral  are  not  super- 
posable. The  difference  is  of  the  same  kind  as  that  between  the  gloves 
of  the  right  and  left  hand. 

Two  non-superposable  rectangular  trihedrals  become  superposable  upon 
reversing  one  (or  all  three)  of  the  axes  of  either  one. 


XIII,  §  297]  QUADRIC  SURFACES  265 

296.  The  fact  that  the  nine  direction  cosines  are  connected  by  six  rela- 
tions (§  292)  suggests  that  it  must  be  possible  to  determine  the  position  of 
the  new  trihedral  with  respect  to  the  old  by  only  three  angles.  As  such 
we  may  take,  in  the  case  of  superposable  trihedrals,  the  angles  6,  0,  ^, 
marked  in  Fig.  135,  which  are  known  as  Eulefs  angles. 

The  figure  shows  the  intersections  of  the  two  trihedrals  with  a  sphere 
of  radius  1  described  about  the  origin  as  center.  If  OiV is  the  intersection 
of  the  planes  Oxy  and  Ox'y',  Euler's  angles  are  defined  as 

d  =  zOz\     (t>  =  NOx',    xfy  =  xON. 

The  line  ON  is  called  the  line  of  nodes,  or  the  nodal  line. 

Imagine  the  new  trihedral  Ox'y'z'  initially  coincident  with  the  old 
trihedral  Oxyz,  in  direction  and  sense.  Now  turn  the  new  trihedral 
about  Oz  in  the  positive  (counterclockwise)  sense  until  Ox'  coincides  with 
the  assumed  positive  sense  of  the  nodal  line  ON;  the  amount  of  this 
rotation  gives  the  angle  \p.  Next  turn  the  new  trihedral  about  ON  in  the 
positive  sense  until  the  plane  Ox'y'  assumes  its  final  position  ;  this  gives 
the  angle  d  as  the  angle  between  the  planes  Oxy  and  Ox'y',  or  the  angle 
zOz'  between  their  normals.  Finally  a  rotation  of  the  new  trihedral 
about  the  axis  Oz',  which  has  reached  its  final  position,  in  the  positive 
sense  until  Ox'  assumes  its  final  position,  determines  the  angle  0. 

297.  The  relations  between  the  nine  direction  cosines  and  the  three 
angles  of  Euler  are  readily  found  from  Fig.  135  by  applying  the  fundamen 
tal  formula  of  spherical  trigonometry  cos  c  =  cos  a  cos  b  +  sin  a  sin  b  cos  y 
successively  to  the  spherical  triangles 

xNx',    xNy',     xNz\ 
yNx',     yNy',     yNz', 

zNx',     zNy',     zNz'. 
We  find  in  this  way  : 

l\  =  cos  ^p  cos  0  —  sin  i/'  sin  0  cos  0, 
mi  =  sin  ^  cos  0  -f  cos  V'  sin  0  cos  6, 
Wi  =  sin  0  sin  d, 

I2—  —  cos  0  sin  0  —  sin  0  cos  0  cos  d,  I3  =  sin  0  sin  0, 

m2  =—  sin  0  sin  0  +  cos  0  cos  0  cos  0,  mz  =—  cos  0  sin  0, 

n2  =  cos  0  sin  0,  m  =  cos  0. 


266 

Four  Place  Logarithms 

N 

0 

1 

2 

3 

.|. 

6 

7 

8   9 

12  3   4  5  6 

7  8  9 

10 

11 
12 
13 

14 
15 

16 

17 

18 
19 

0000 

0414 
0792 
1139 

1461 
1761 
2041 

2304 
2553 

2788 

0043 

0453 
0828 
1173 

1492 
1790 
206« 

^330 
2577 
2810 

0086 

0492 
0864 
1206 

1523 

1818 
2095 

2356 
2(X)1 
2833 

0128 

0531 
0899 
1239 

1553 
1847 
2122 

2380 
2625 
2856 

0170 

0212 

0253 

0294  0334  0374 

4  8  12  1  17  21  25 

29  33  37 

05(59 
0934 
1271 

1584 

isTr. 

2148 

2405 
2648 

2878 

0607 
0969 
1303 

1614 
1903 
2175 

2430 
2672 
2900 

0(545 
1004 
1335 

1644 

v.m 

2201 

2455 
2695 
2923 

0682 
1038 
1367 

1673 
1959 
2227 

2480 
2718 
2^>45 

0719 
1072 
1399 

1703 
1987 
??53 

2504 
2742 
2967 

0755 
1106 
1430 

1732 
2014 
2279 

252<) 
2765 
2989 

4  8  11 
3  7  10 
3  6  10 

3  6  9 
3  6  8 
3  5  8 

2  5  7 
2  5  7 

2  4  7 

15  19  23 
14  17  21 
13  16  19 

12  15  18 
11  14  17 
11 13  16 

10  12  15 
9  1214 
9  1113 

26  30S4 
24  28  31 
23  26  29 

2124  27 
20  22  25 
18  21  24 

17  20  22 
16  19  21 
16  18  20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

2  4  6 

8  1113 

16  17  19 

21 
22 
23 

24 
25 

26 

27 
28 
29 

30 

3222 
3424 
3617 

3802 
3979 
4150 

4314 
4472 
4624 

4771 

3243 
3444 
3636 

3820 
3997 
4166 

4330 

4487 
4639 

3263 
34(>4 
3655 

3838 
4014 
4183 

4346 
4502 
4654 

3284 
3483 
3674 

3856 
4031 
4200 

4362 
4518 

m>9 

3304 
3502 
3692 

3874 
4048 
4216 

4378 
4533 
4683 

3324 
3522 
3711 

3892 
4065 
4232 

4393 
4548 
4698 

3346 
3541 
3729 

3909 
4082 
4249 

4409 
4564 
4713 

a365 
3560 
3747 

3927 
4099 
4265 

4425 
4579 
4728 

3385 
3579 
3766 

3945 
4116 
4281 

4440 
451  »4 
4742 

»404 
3598 
3784 

3962 
413,3 
4298 

445(5 
4(509 
4757 

2  4  6 
2  4  6 
2  4  6 

2  4  6 
2  4  6 
2  3  5 

2  3  6 
2  3  5 
13  4 

8  10  12 
8  10  12 
7  9  11 

7  9  11 
7  9  10 
7  8  10 

6  8  9 
6  8  9 
6  7  9 

14  16  18 
14  16  17 
13  15  17 

12  14  10 
12  14  16 
11  13  15 

11  12  14 
11  12  14 
10  12  13 

4786 

4800 

4814 

4829  4843 

4857 

4871 

4886 

4900 

13  4 

6  7  9 

10  11  13 

31 
32 
33 

34 
35 
36 

37 
38 
39 

4914 
5051 
5185 

6315 
5441 
556;^ 

5682 
5798 
6911 

4928 
5066 
5198 

5328 
5453 
5575 

5694 
5809 
6922 

4942 
5079 
5211 

5340 
6465 
5587 

5706 
5821 
6933 

4965 
5092 
5224 

5353 

5478 
5599 

5717 
6832 
5944 

4969  4983 
5105  5119 
5237  5250 

5366  5378 
5490  5502 
5611  5623 

5729  5740 

5843  5855 
5955  59(3() 

4997 
5132 
5263 

5391 
5514 
5635 

5752 

r>m) 

5977 

5011 
5145 
5276 

5403 
5527 
5&47 

5763 
5877 
5988 

5024 
5159 
5289 

5416 
5539 
6658 

6775 
5888 
5999 

5038 
5172 
5302 

5428 
5551 
5670 

5786 
6899 
6010 

1  3  4 
13  4 
13  4 

1  2  4 
1  2  4 
12  4 

1  2  4 
1  2  3 
1  2  3 

5  7  8 
5  7  8 
5  7  8 

5  6  8 
5  6  7 
5  6  7 

5  6  7 
5  6  7 
4  6  7 

10  11  12 
911  12 
91112 

910  11 
910  11 
810  11 

8  9  11 
8  9  10 
8  910 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6086 

6096 

6107 

6117 

12  3 

4  5  6 

8  9  10 

41 

42 
43 

44 
45 

46 

47 
48 
49 

6128 
6232 
6335 

6435 
6632 
6628 

6721 
6812 
6902 

6138 
6243 
6345 

6444 
6542 
6637 

6730 
6821 
6911 

6149 
6253 
6355 

6454 
(;551 
6646 

6739 
6830 
6920 

6160 
6263 
6365 

6464 

6656 

6749 
6839 
6928 

6170 
6274 
6375 

6474 
6571 
6665 

6758 
6*48 
6937 

6180 
62S4 
a385 

6484 
6580 
6675 

6767 
6857 
6W6 

6191 
6294 
6395 

&493 
6;-)90 
6684 

6776 
6866 
6955 

6201 
6304 
6405 

6503 
6599 
6693 

6785 
6875 
6964 

6212 
6314 
6415 

()513 
6609 
6702 

6794 
6884 
6972 

6222 
6325 
6425 

6522 
6618 
6712 

6803 
6893 
6981 

1  2  3 
12  3 
12  3 

12  3 
1  2  3 
12  3 

12  3 
12  3 
12  3 

4  5  6 
4  5  6 
4  5  6 

4  5  6 
4  5  6 
4  5  6 

4  5  6 
4  5  6 
4  4  6 

7  8  9 
7  8  9 
7  8  9 

7  8  9 
7  8  9 
7  7  8 

7  7  8 
7  7  8 
6  7  8 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7060 

7059 

7067 

12  3 

3  4  6 

6  7  8 

51 
52 
53 

54 

7076 
7160 
7243 

7324 

70&4 
7168 
7251 

7332 

7093 
7177 
7269 

7340 

7101 
7185 
7267 

7348 

7110 
7193 
7275 

7356 

7118 
7202 
7284 

7364 

7126 
7210 
7292 

7372 

7135 
7218 
7300 

7380 

7143 
7226 
7308 

7388 

7152 
7235 
7316 

Tim 

12  3 
1  2  3 
1  2  2 

1  2  2 

3  4  5 
3  4  5 
3  4  5 

3  4  5 

6  7  8 
6  7  7 
6  6  7 

6  6  7 

N 

0 

1 

2 

8 

4 

6 

6 

7 

8 

9 

1  2  2 

4  5  6 

7  8  9 

The  proportional  parts  are  stated  in  full  for  every  tenth  at  the  right-hand  side. 
The  logarithm  of  any  number  of  four  significant  figures  can  bo  read  directly  by  add- 


Four  Place  Logarithms 

267 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

12   3 

4   5   6 

7   8   9 

55 

56 

57 
58 
59 

60 

7404 
7482 

7559 
7634 
7709 

7782 

7412 

7490 

7566 
7642 
7716 

7789 

7419 

7497 

7574 
7649 
7723 

7427 
7505 

7582 
7657 
7731 

7435 
7513 

7589 
7664 

7738 

7443 
7520 

7597 
7672 
7745 

7451 

7528 

7604 
7679 
7752 

7459 
7536 

7612 

7686 
7760 

7466 
7543 

7619 
7694 
7767 

7474 
7551 

7(527 
7701 
7774 

12    2 
12    2 

112 
112 
112 

3   4   5 
3,  4   5 

3   4    5 
3    4   4 
3   4    4 

5  6  7 
5   6   7 

5  6  7 
5  6  7 
5    6    7 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

112 

3    4    4 

5    6    6 

61 
62 
63 

64 
65 

66 

67 
68 
69 

7853 
7924 
7993 

8062 
8129 
8195 

8261 
8325 
8388 

7860 
7931 
8000 

8069 
8136 
8202 

8267 
8331 
8395 

7868 
7938 
8007 

8075 
8142 
8209 

8274 
8338 
8401 

7875 
7945 
8014 

8082 
8149 
8215 

8280 
8344 
8407 

7882 
7952 
8021 

8089 
8156 
8222 

8287 
8351 
8414 

7889 
7959 
8028 

8096 
8162 
8228 

8293 
8357 
8420 

7896 
7966 
8035 

8102 
8169 
8235 

8299 
8363 
8426 

7903 
7973 
8041 

8109 
8176 
8241 

8306 
8370 
8432 

7910 
7980 
8048 

8116 
8182 
8248 

8312 
8376 
8439 

7917 
7987 
8055 

8122 
8189 
8254 

8319 
8382 
8445 

112 
1    1    2 
112 

112 
112 
112 

112 
112 
112 

3    3    4 
3    3    4 
3   3   4 

3   3   4 
3   3   4 
3   3   4 

3   3   4 
3    3   4 
3    3   4 

5  6  6 
5  5  6 
5   5   6 

5  5  6 
5    5   6 

5  5   6 

6  5  6 
4  5  6 
4    5    6 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

112 

3   3   4 

4   5   6 

71 
72 
73 

74 
75 

76 

77 
78 
79 

8513 
8573 
8633 

8692 
8751 
8808 

8865 
8921 
8976 

8519 
8579 
8639 

8698 
8756 
8814 

8871 
8927 
8982 

8525 
8585 
8645 

8704 
8762 
8820 

8876 
8932 
8987 

8531 
8591 
8651 

8710 

8768 
8825 

8882 
8938 
8993 

8537 
8597 
8657 

8716 

8774 
8831 

8887 
8943 
8998 

8543 
8603 
8663 

8722 
8779 
8837 

8893 
8949 
9004 

8549 
8609 
8669 

8727 
8785 
8842 

8899 
8954 
9009 

8555 
8615 
8675 

8733 
8791 
8848 

8904 
8960 
9015 

8561 
8621 
8681 

8739 
8797 
8854 

8910 
8965 
9020 

8567 
8627 
8686 

8745 
8802 
8859 

8915 
8971 
9025 

112 
112 
112 

112 
112 
112 

1    1    2 
112 
112 

3    3   4 
3    3   4 
2   3   4 

2   3   4 
2    3    3 
2    3   3 

2   3   3 
2    3   3 
2    3    3 

4  5  6 
4  5  6 
4   5    6 

4  6  6 
4   5    5 

4   4   5 

4  4  5 
4  4  5 
4    4    5 

80 

^)031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

112 

2    3    3 

4    4   5 

81 
82 
83 

84 
85 

86 

87 
88 
89 

90a5 
9138 
9191 

9243 
9294 
9345 

9395 
9445 
9494 

fX)90 
9143 
9196 

9248 
9299 
9350 

9400 
9450 
9499 

9096 
9149 
9201 

9253 
9304 
9355 

9405 
9455 
9504 

9101 
9154 
9206 

9258 
9309 
9360 

9410 
9460 
9509 

910fj 
9159 
9212 

9263 
9315 
9365 

9415 
9465 
9513 

9112 
9165 
9217 

9269 
9320 
9370 

9420 
9469 
9518 

9117 
9170 
9222 

9274 
9325 
9375 

9425 
9474 
9523 

9122 
9175 
9227 

9279 
9330 
9380 

9430 
9479 
9528 

9128 
9180 
9232 

9284 
9335 
9385 

9435 
9484 
9533 

9133 
9186 
9238 

9289 
9340 
9390 

9440 
9489 
9538 

112 
1    1    2 
1    1    2 

112 
1    1    2 
112 

112 
0    1    1 
0    1    1 

2    3    3 
2   3    3 
2   3   3 

2    3    3 
2    3   3 
2    3   3 

2    3   3 
2    2    3 

2    2    3 

4  4  5 
4  4  5 
44    5 

4  4  5 
4  4  5 
4   4    5 

4  4  5 
3  4  4 
3    4    4 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

0    1    1 

2    2    3 

3    4    4 

91 
92 
93 

94 
95 

96 

97 
98 
99 

9590 
9638 
9685 

9731 
9777 
9823 

9868 
9912 
9956 

9595 
9643 
9689 

9736 
9782 
9827 

9872 
9917 
9961 

9600 
9647 
9694 

9741 
9786 
9832 

9877 
9<)21 
99(i5 

9605 
9652 
9699 

9745 
9791 
9836 

9881 
9926 
9969 

9609 
9657 
9703 

9750 
9795 
9841 

9886 
9930 
9974 

9614 
9661 
9708 

9754 
9800 
9845 

9890 
9934 
9978 

9619 
9666 
9713 

9759 
9805 
9850 

9894 
9939 
9983 

9624 

mil 

9717 

9763 
9809 
9854 

9899 
9943 
9987 

9628 
9675 
9722 

9768 
9814 
9859 

t)903 
9948 
9991 

9633 
9(i80 
9727 

9773 
9818 
9863 

9908 
9952 
9996 

Oil 
Oil 
Oil 

Oil 
0    1    1 
0    1    1 

Oil 
Oil 
Oil 

2    2    3 
2    2    3 
2   2   3 

2    2    3 
2   2    3 
2   2    3 

2    2    3 
2    2    3 
2    2    3 

3  4  4 
3  4  4 
3    4   4 

3  4  4 
3  4  4 
3   4   4 

3  4  4 
3  3  4 
3    3    4 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1    2    3 

4    5    6 

7    8    9 

ing  the  proportional  part  corresponding  to  the  fourth  figure  to  the  tabular  number 
corresponding  to  tbo  first  three  figures.    Thero  may  be  an  error  of  1  in  the  last  place. 


268 


Four  Place  Trigonometric  Functions 


[Charactoristics  of  Lof?arithins  orn 

tted- 

determine  by  the  usual  i 

-ule  from  the  value] 

Uadians 

1  )egrees 

Si 

ME 

Tangext 

C!oTANGENT 

Cosi 

NE 

Value 

Lo?io 

Value 

Logio 

Value 

Logio 

Value 

Lopio 

.0000 

0°00' 

.0000 

.0000 

l.OCKX) 

.0000 
.0000 

90°  00' 

50 

1.5708 
1.5079 

!0029 

10 

.0029 

.4637 

!0029 

.4637 

343.77 

.5363 

I'.OOOO 

.0058 

20 

.0058 

.7(>48 

.0058 

.7648 

171.89 

.2352 

1.0000 

.0000 

40 

1.5650 

.0087 

30 

.0087 

.9408 

.0087 

.i)409 

114.69 

.0591 

1.0000 

.0000 

30 

1.5(521 

.0116 

40 

.0116 

.0(558 

.0116 

.0658 

85.940 

.9^42 

.9995) 

.0000 

20 

1.5592 

.0U5 

50 

.0145 

.1627 

.0145 

.1627 

68.750 

.8373 

.9999 

.0000 

10 

1.5563 

.0175 

1°00' 

.0175 

.2419 

.0175 

.2419 

57.290 

.7581 

.9998 

.99^)0 

89°  00' 

1.5533 

.0204 

10 

.0204 

.3088 

.0204 

.3089 

49.104 

.6911 

.9998 

.fn>9<) 

50 

1.5504 

.0233 

20 

.0233 

.3668 

.0233 

.36(59 

42.964 

.6331 

.9i)97 

.99*.><) 

40 

1.5475 

.0202 

30 

.0262 

.4179 

.02(52 

.4181 

38.188 

.5819 

.99i)7 

.91HHJ 

30 

1.5446 

.0291 

40 

.0291 

.4637 

.0291 

.4638 

34.368 

.5362 

.9996 

.9<)98 

20 

1.5417 

.0320 

50 

.0320 

.5050 

.0320 

.6053 

31.242 

.4947 

.9995 

.ims 

10 

1.5388 

.0349 

2°  00' 

.0^49 

.5428 

.0349 

.5431 

28.636 

.4569 

.9994 

.9997 

88°  00' 

1.5359 

.0378 

10 

.0378 

.5776 

.0378 

.5779 

26.4.32 

.4221 

.9993 

.99<)7 

50 

1.53.30 

.0407 

20 

.0407 

.6097 

.0407 

.6101 

24.542 

.3899 

.9{m 

.9^>96 

40 

1.5301 

.0436 

30 

.0436 

Ami 

.0437 

.6401 

22.904 

.3599 

.9990 

.^y^m 

30 

1.5272 

.0465 

40 

.0465 

.(5677 

.0466 

.6(582 

21.470 

.3318 

.9989 

.9995 

20 

1.5243 

.0495 

50 

.0494 

.(5940 

.0495 

.6945 

20.20(5 

.3055 

.9988 

.9995 

10 

1.5213 

.0524 

3°  00' 

.0523 

.7188 

.0524 

.7194 

19.081 

.2806 

.9986 

.9994 

87° 00' 

1.5184 

.0553 

10 

.0552 

.7423 

.0553 

.7429 

18.075 

.2571 

.9985 

.V.W3 

50 

1.5155 

.0582 

20 

.0581 

.7645 

.0582 

.7(552 

17.1(59 

.2^48 

.9i)83 

.9993 

40 

1.5126 

.0611 

30 

.0(510 

.7a')7 

.0612 

.78(55 

16..350 

.2i;i5 

.9981 

.m\2 

30 

1.5097 

.0(U0 

40 

.0640 

.8059 

.0(541 

.80(57 

15.(505 

.193:i 

.9980 

.mn 

20 

1.5068 

.0669 

50 

.0<}<39 

.8251 

.0(570 

.8261 

14.924 

.1739 

.9978 

.9f»90 

10 

1.5039 

.0698 

4°  00' 

.0(598 

.8436 

.0699 

.8446 

14..301 

.1554 

.9976 

.9989 

86°  00' 

1.5010 

.0727 

10 

.0727 

.8613 

.0729 

.8(524 

13.727 

.1376 

.9974 

.9!>S9 

50 

1.4981 

.0756 

20 

.0756 

.8783 

.0758 

.8795 

13.197 

.1205 

.9971 

.9988 

40 

1.4952 

.0785 

30 

.0785 

.8946 

.0787 

.8960 

12.706 

.1040 

.9969 

.9987 

30 

1.4923 

.0814 

40 

.0814 

.9104 

.0816 

.9118 

12.251 

.0882 

.9967 

.9986 

20 

1.4893 

.0844 

50 

.0M3 

.9256 

.0*46 

.9272 

11.826 

.0728 

.9964 

.9985 

10 

1.4864 

.0873 

6°  00' 

.0872 

.9403 

.0875 

.9420 

11.430 

.0580 

.9962 

SY.)H-A 

85° 00' 

1.4835 

.0902 

10 

.mK)l 

.9545 

.0904 

.9m:i 

11.059 

.0437 

.{)959 

A¥M'2 

50 

1.4806 

.0931 

20 

.0929 

sm2 

.09:i4 

.9701 

10.712 

.02^9 

.9957 

.9<t81 

40 

1.4777 

.OiXJO 

30 

.0958 

.9816 

.0963 

.98:3(5 

10..'385 

.0164 

.*«»54 

.9980 

30 

1.4748 

.0989 

40 

.0987 

.9<H5 

.09<>2 

.99(56 

10.078 

.0034 

.9951 

.9«>79 

20 

1.4719 

.1018 

50 

.1016 

.0070 

.1022 

.0093 

9.7882 

.9907 

.9948 

.i>977 

10 

1.4690 

.1047 

6°  00' 

.1045 

.0192 

.ia5i 

.0216 

9.5144 

.9784 

.9946 

.9976 

84°  00' 

1.4661 

.1076 

10 

.1074 

.0311 

.1080 

.o:»6 

9.2553 

.9(5(54 

.9942 

.9<)75 

50 

1.4632 

.1105 

20 

.1103 

.0426 

.1110 

.0453 

9.0098 

.9547 

.9939 

.9973 

40 

1.4603 

.1134 

30 

.1132 

.0539 

.1139 

.0567 

8.77(59 

.94:« 

.9936 

.9972 

30 

1.4573 

.1164 

40 

.11(51 

.0648 

.11(59 

.0678 

8.5555 

.9322 

.9i)32 

.9»»71 

20 

1.4544 

.1193 

50 

.1190 

.0755 

.1198 

.0786 

8.3450 

.9214 

.9929 

.9969 

10 

1.4516 

.1222 

7°  00' 

.1219 

.0859 

.1228 

.0891 

8.1443 

.9109 

.9925 

.99(58 

83°  00' 

1.4486 

.1251 

10 

.1248 

.0961 

.1257 

.0995 

7.9530 

.9005 

.9i>22 

.9966 

60 

1.4457 

.1280 

20 

.1276 

.10(50 

.1287 

.1096 

7.7704 

.8904 

.9918 

.9964 

40 

1.4428 

.1309 

30 

.1305 

.1157 

.1517 

.1194 

7.5958 

.8806 

.9914 

.9963 

30 

1.4399 

.1338 

40 

.1334 

.1252 

.1346 

.1291 

7.4287 

.8709 

.9911 

.9961 

20 

1.4370 

.13(>7 

50 

.1363 

.1345 

.1376 

.1385 

7.2687 

.8615 

.9907 

.9iW 

10 

1.4»41 

.1396 

8°  00' 

.1392 

.1436 

.1405 

.1478 

7.1154 

.8522 

.9903 

.9958 

82°  00' 

1.4312 

.1425 

10 

.1421 

.152,5 

.1435 

.1569 

6.9682 

.8431 

.9899 

.995(5 

50 

1.4283 

.1454 

20 

.1449 

.1612 

.1465 

.1(558 

6.8269 

.8342 

.aS94 

.9954 

40 

1.4254 

.148-4 

30 

.1478 

.1697 

.1495 

.1745 

6.6912 

.82.55 

.9890 

.{K)52 

30 

1.4224 

.1513 

40 

.1507 

.1781 

.1524 

.ia3i 

6.5(506 

.8169 

.988(5 

.99.50 

20 

1.4195 

.1542 

50 

.1536 

.1863 

.1554 

.1915 

6.4348 

.8085 

.9881 

.9948 

10 

1.4166 

.1571 

9°  00' 

.1564 

.1943 

.1584 

.1997 

6.3138 

.8003 

.9877 

.9946 

81°  00' 

1.4137 

Value 

Log,o 

Value 

Lopio 

Value 

Logio 

Value 

Logio 

Degrees 

Radians 

Cosine 

Cotangent 

Tangent 

Sink 

Four  Place  Trigonometric  Functions 


269 


[Characteristics  of  Logarith 

ms  orai 

tted  — 

leterraine  by  the  usual  rule  from  the  value] 

Radians 

Degeees 

Sine 

Tangent 

Cotangent 

Cosine 

V^alue  Logjo 

Value 

i-Ogio 

Value 

Logio 

Value 

Logio 

.1571 

9°  00' 

.1564  .1943 

.1584 

.1997 

6.3138 

.8003 

.9877 

.9946 

81°  00' 

1.413? 

.1600 

10 

.1593  .2022 

.1614 

.2078 

6.1970 

.7922 

.9872 

.9944 

50 

1.4108 

.1(529 

20 

.1622  .2100 

.1644 

.2158 

6.0844 

.7842 

.98(58 

.9942 

40 

1.4079 

.1658 

30 

.1650  .217() 

.1673 

.2236 

5.9758 

.7764 

.9863 

.9940 

30 

1.4050 

.1687 

40 

.1679  .2251 

.1703 

.2313 

5.8708 

.7687 

.9858 

.9938 

20 

1.4021 

.1716 

50 

.1708  .2324 

.1733 

.2389 

5.7694 

.7611 

.9853 

.t)936 

10 

1.3992 

.1745 

10°  00' 

.1736   .2397 

.1763 

.2463 

5.6713 

.7537 

.9848 

.9934 

80°  00' 

1.3963 

.1774 

10 

.1765  .2468 

.1793 

.2536 

5.5764 

.7464 

.9843 

.9931 

50 

1.3934 

.1804 

20 

.1794  .2538 

.1823 

.2609 

5.4845 

.7391 

.9838 

.9929 

40 

1.3904 

.1833 

30 

.1822  .2606 

.1853 

.2680 

5.3955 

.7320 

.9833 

.9927 

30 

1.3875 

.1862 

40 

.J851  .2674 

.1883 

.2750 

5.3093 

.7250 

.9827 

.9924 

20 

1.3846 

.1891 

50 

.1880  .2740 

.1914 

.2819 

5.2257 

.7181 

.9822 

.9922 

10 

1.3817 

.1920 

11°00' 

.1908  .2806 

.1944 

.2887 

5.1446 

.7113 

.9816 

.9919 

79°  00' 

1.3788 

.1949 

10 

.1937  .2870 

.1974 

.2953 

5.06.58 

.7047 

.9811 

.9917 

50 

1..3759 

.1978 

20 

.1965  .2934 

.2004 

.3020 

4.98<)4 

.6980 

.9805 

.9914 

40 

1.3730 

.2007 

30 

.1994  .25)97 

.2035 

.3085 

4.9152 

.6915 

.9799 

.9912 

30 

1..3701 

.2036 

40 

.2022  .3058 

.2065 

.3149 

4.8430 

.6851 

.9793 

.9909 

20 

1.3672 

.2065 

50 

.2051  .3119 

.2095 

.3212 

4.7729 

.6788 

.9787 

.9907 

10 

1.3(343 

.2094 

12<'00' 

.2079  .3179 

.2126 

.3275 

4.7046 

.6725 

.9781 

.9904 

78°  00' 

1.3614 

.2123 

10 

.2108  .3238 

.2156 

.3336 

4.6382 

.6664 

.9775 

.9901 

50 

1.3584 

.2153 

20 

.2136  .32% 

.2186 

.3397 

4.5736 

.6603 

.9769 

.9899 

40 

1..3555 

.2182 

30 

.2164  .3353 

.2217 

.3458 

4.5107 

.6542 

.9763 

.9896 

30 

1.3526 

.2211 

40 

.2193  .3410 

.2247 

.3517 

4.4494 

.6483 

.9757 

.9893 

20 

1.3497 

.2240 

50 

.2221  .3466 

.2278 

.3576 

4.3897 

.6424 

.9750 

.9890 

10 

1.3468 

.2269 

13°  00' 

.2250  .3521 

.2309 

.3(534 

4.3315 

.6366 

.9744 

.9887 

77°  00' 

1.3439 

.2298 

10 

.2278  .3575 

.2339 

.3691 

4.2747 

.6309 

.9737 

.9884 

50 

1.3410 

.2327 

20 

.2306  .3629 

.2370 

.3748 

4.2193 

.6252 

.9730 

.9881 

40 

1.3381 

.2356 

30 

.2334  .3682 

.2401 

.3804 

4.1653 

.619(5 

.9724 

.9878 

30 

1.3352 

.2385 

40 

.2;^)3  .3734 

.24.32 

.3859 

4.1126 

.6141 

.9717 

.9875 

20 

1.3323 

.2414 

50 

.2391  .3786 

.2462 

.3914 

4.0<ill 

.6086 

.9710 

.9872 

10 

1.3294 

.2443 

14°  00' 

.2419  .3837 

.2493 

.3968 

4.0108 

.6032 

.9703 

.98(39 

76°  00' 

1.3265 

.2473 

10 

.2447  .3887 

.2524 

.4021 

3.9(517 

.5979 

.961K5 

.98(56 

50 

1..3235 

.2502 

20 

.2476  .3937 

.2555 

.4074 

3.9136 

.5926 

.9689 

.9863 

40 

1.3206 

.2531 

30 

.2504  .3986 

.2586 

.4127 

3.8667 

.5873 

.9(581 

.9859 

30 

1.3177 

.2560 

40 

.2532  .4035 

.2617 

.4178 

3.8208 

.5822 

.9674 

.985(5 

20 

1.3148 

.2589 

50 

.2560  .4083 

.2648 

.4230 

3.77(30 

.5770 

.fH367 

.9853 

10 

1.3119 

.2618 

15°00' 

.2588  .4130 

.2679 

.4281 

3.7321 

.5719 

.9659 

.9849 

75°  00' 

1.30^)0 

.2647 

10 

.2616  .4177 

.2711 

.4331 

3.6891 

.5669 

.9(352 

.9846 

50 

1.3061 

.2676 

20 

.2641  .4223 

.2742 

.4381 

3.6470 

.5619 

.9644 

.9843 

40 

1.3032 

.2705 

30 

.2672  .4269 

.2773 

.4430 

3.6059 

.5570 

.9(336 

.9839 

30 

1.3003 

.2734 

40 

.2700  .4314 

.2805 

.4479 

3.5656 

.5521 

.9628 

.9836 

20 

1.2974 

.2763 

50 

.2728  .4359 

.2836 

.4527 

3.5261 

.5473 

.9621 

.9832 

10 

1.2945 

.2793 

16°  00' 

.2756  .4403 

.2867 

.4575 

3.4874 

.5425 

.9613 

.9828 

74°  00' 

1.2915 

.2822 

10 

.2784  .4447 

.2899 

.4622 

3.4495 

.5378 

.%05 

.9825 

50 

1.2886 

.2851 

20 

.2812  .4491 

.2931 

.4669 

3.4124 

.5331 

.9596 

.9821 

40 

1.2857 

.2880 

30 

.2840  .45.33 

.2962 

.4716 

3.3759 

.5284 

.9588 

.9817 

30 

1.2828 

.2909 

40 

.2868  .4576 

.2994 

.4762 

3.3402 

.52.38 

.9580 

.9814 

20 

1.2799 

.2938 

50 

.2896  .4618 

.3026 

.4808 

3.3052 

.5192 

.9572 

.9810 

10 

1.2770 

.2967 

17°  00' 

.2924  .4659 

.3057 

.4853 

3.2709 

.5147 

.9563 

.9806 

73°  00' 

1.2741 

.2996 

10 

.2952  .4700 

.3089 

.4898 

3.2371 

.5102 

.9555 

.9802 

50 

1.2712 

.3025 

20 

.2979  .4741 

.3121 

.4943 

3.2041 

.5057 

.9546 

.9798 

40 

1.2683 

.3054 

30 

.3007  .4781 

.3153 

.4987 

3.1716 

.5013 

.9537 

.9794 

30 

1.2654 

.3083 

40 

.3035  .4821 

.3185 

.5031 

3.1397 

.4969 

.9528 

.9790 

20 

1.2625 

.3113 

50 

.3062  .4861 

.3217 

.5075 

3.1084 

.4925 

.9520 

.9786 

10 

1.2595 

.3142 

18°  00' 

.3090  Aim 

.3249 

.5118 

3.0777 

.4882 

.9511 

.9782 

72°  00' 

1.2.566 

Value  Logio 

Value 

Login 

Value 

Logio 

Value 

Logio 

Degrees 

Radians 

Cosine 

Cotangent 

Tangent 

Sine 

270 


Four  Place  Trigonometric  Functions 


[Characteristics  of  Logarithms  omitted  — 

determine  by  the  usual  rule  from  the  value] 

Radians 

Dbobees 

Sink 

Tangent 

Cotangent 

Cosine  . 

Value 

Logio 

Value 

Logio 

Value 

Logio 

Value  Logio 

.3142 

18° 00' 

.3090 

.4900 

.3249 

.5118 

3.0777 

.4882 

.9511  .9782 

72°  00' 

1.2566 

.3171 

10 

.3118 

.4939 

.3281 

.5161 

3.0475 

.4839 

.9502  .9778 

50 

1.2537 

.3200 

20 

.3145 

.4977 

.3314 

.5203 

3.0178 

.4797 

.9492  .i)774 

40 

1.2508 

.3229 

30 

.3173 

.5015 

.3346 

.5245 

2.9887 

.4755 

.9483  .9770 

30 

1.2479 

.3258 

40 

.3201 

.5052 

.3378 

.5287 

2.9600 

.4713 

.9474  .9765 

20 

1.2450 

.3287 

50 

.3228 

.5090 

.3411 

.5329 

2.9319 

.4671 

.9465  .9761 

10 

1.2421 

.3316 

19°  00' 

.3256 

.5126 

.3443 

.5370 

2.9042 

.4630 

.9455  .9757 

71°  00' 

1.2392 

.3345 

10 

.3283 

.5163 

.3476 

.5411 

2.8770 

.4589 

.9446  .9752 

50 

1.2.363 

.3374 

20 

.3311 

.5U)9 

.3508 

.5451 

2.8502 

.4549 

.9436  .9748 

40 

1.2334 

.»i03 

30 

.3338 

.5235 

.3541 

.5491 

2.8239 

.4509 

.9426  .9743 

30 

1.2305 

.3432 

40 

.3365 

.5270 

.3574 

.5531 

2.7980 

.4469 

.9417  .9739 

20 

1.2275 

.3462 

50 

.3393 

.5306 

.3607 

.5571 

2.7725 

.4429 

.9407  .9734 

10 

1.2246 

.3491 

20°  00' 

.3420 

.5341 

.3640 

.5611 

2.7475 

.4389 

.9397  .9730 

70°  00' 

1.2217 

.3520 

10 

.3448 

.5375 

.3673 

.5650 

2.7228 

.4350 

.9387  .9725 

50 

1.2188 

.3549 

20 

.3475 

.5409 

.3706 

.5689 

2.6985 

.4311 

.9377  .9721 

40 

1.2159 

.3578 

30 

.3502 

.5443 

.3739 

.5727 

2.6746 

.4273 

.9367  .9716 

30 

1.2130 

.3607 

40 

.3529 

.5477 

.3772 

.5766 

2.6511 

.4234 

.9356  .9711 

20 

1.2101 

.3636 

50 

.3557 

.5510 

.3805 

.5804 

2.6279 

.4196 

.9346  .9706 

10 

1.2072 

.3665 

21°  00' 

.3584 

.5543 

.3839 

.5842 

2.6051 

.4158 

.9336  .9702 

69° 00' 

1.2043 

.3694 

10 

.3611 

.5576 

.3872 

.5879 

2.5826 

.4121 

.9.325  .9697 

60 

1.2014 

.3723 

20 

.3638 

.5609 

.3906 

.5917 

2.5605 

.4083 

.9315  .9692 

40 

1.1985 

.3752 

30 

.3665 

.5641 

.3939 

.5954 

2.5386 

.4046 

.9304  .9687 

30 

1.1956 

.3782 

40 

.3692 

.5673 

.3973 

.mn 

2.5172 

.4009 

.9293  .9682 

20 

1.1926 

.3811 

50 

.3719 

.5704 

.4006 

.6028 

2.4960 

.3972 

.9283  .9677 

10 

1.1897 

.3840 

22°  00' 

.3746 

.5736 

.4040 

.6064 

2.4751 

.3936 

.9272  .9672 

68°  00' 

1.1868 

.3869 

10 

.3773 

.5767 

.4074 

.6100 

2.4545 

.3900 

.9261  .9667 

50 

1.1839 

.3898 

20 

.3800 

.5798 

.4108 

.6136 

2.4342 

.3864 

.9250  .9661 

40 

1.1810 

.3i)27 

30 

.3827 

.5828 

.4142 

.6172 

2.4142 

.3828 

.i)239  .9656 

30 

1.1781 

.3956 

40 

.3854 

,5859 

.4176 

.()208 

2.31^ 

.37i)2 

.9228  .9651 

20 

1.1752 

.3985 

50 

.3881 

.5889 

.4210 

.0243 

2.3750 

.3757 

.9216  .9646 

10 

1.1723 

.4014 

23°  00' 

.3907 

.5919 

.4245 

.6279 

2.a559 

.3721 

.9205  .9640 

67°  00' 

1.1094 

.4043 

10 

.S9M 

.5948 

.4279 

.6314 

2.;i:3()9 

.3686 

.91i>4  .9635 

50 

1.1665 

.4072 

20 

.3961 

.5978 

.4314 

.o:j48 

2.3183 

.3652 

.9182  .9()2f) 

40 

1.163(5 

.4102 

30 

.3987 

.6007 

.4348 

.6383 

2.2^)98 

.3617 

.9171  .9624 

30 

1.1606 

.4131 

40 

.4014 

.6036 

.4383 

.6417 

2.2817 

.3583 

.9159  .9618 

20 

1.1577 

.4160 

50 

.4041 

.6065 

.4417 

.6452 

2.2637 

.3548 

.9147  .9613 

10 

1.1548 

.4189 

24°  00' 

.4067 

.6093 

.4452 

.6486 

2.2460 

.3514 

.9135  .9607 

66° 00' 

1.1519 

.4218 

10 

.4094 

.6121 

.4487 

.6520 

2.2286 

.3480 

.9124  .9602 

50 

1.1490 

.4247 

20 

.4120 

.6149 

.45-22 

.()553 

2.2113 

.3447 

.9112  .9596 

40 

1.1461 

.4276 

30 

.4147 

.6177 

.4557 

.6587 

2.1943 

.3413 

.9100  .9590 

30 

1.1432 

.4305 

40 

.4173 

.6206 

.45i)2 

.6620 

2.1775 

.3380 

.9088  .9584 

20 

1.1403 

.4334 

50 

.4200 

.6232 

.4628 

.6654 

2.1609 

.3346 

.9075  .9579 

10 

1.1374 

.4363 

26°  00' 

.4226 

.6259 

.4663 

.6687 

2.1445 

.3313 

.9063  .9573 

66°  00' 

1.1345 

.4392 

10 

.4253 

.628(5 

.4699 

.6720 

2.1283 

.3280 

.tK)51  .95<57 

50 

1.1316 

.4422 

20 

.4279 

.6313 

AIM 

.67r,2 

2.1123 

.3248 

.9038  .9561 

40 

1.1286 

.4451 

30 

.4305 

.6340 

.4770 

.6785 

2.0<K>5 

.3215 

.9026  .9555 

30 

1.1267 

.4480 

40 

.4331 

.6.36<i 

.4806 

.6817 

2.0809 

.3183 

.9013  .a549 

20 

1.1228 

.4509 

50 

.4358 

.6392 

.4S41 

.68.50 

2.0655 

.3150 

.9001  .9543 

10 

1.1199 

.4538 

26°  00' 

.4384 

.6418 

.4877 

.6882 

2.0503 

.3118 

.8988  .9537 

64°  00' 

1.1170 

.4567 

10 

.4410 

.6444 

.4913 

.6914 

2.o;r)3 

.3086 

.85)75  .a')30 

50 

1.1141 

.4596 

20 

.4436 

.6470 

.4950 

.<)1H6 

2.0204 

.3054 

.8962  .9524 

40 

1.1112 

.4625 

30 

.4462 

.6495 

.4986 

.6977 

2.0057 

.3023 

.8949  .9518 

30 

1.1083 

.4654 

40 

.4488 

.6521 

.5022 

.7009 

1.9912 

.2991 

.S9m   .9512 

20 

1.1054 

.4683 

50 

.4514 

.6540 

.5059 

.7040 

1.9768 

.2960 

.8923  .9505 

10 

1.1025 

.4712 

27°  00' 

.4540 

.6570 

.5095 

.7072 

1.9626 

.2928 

.8910  .9499 

63°  00' 

1.0996 

Value 

Logio 

Value 

Logio 

Value 

Logio 

Value  Logio 

Degbebs 

Radians 

Cosine 

Cotangent 

Tangent 

Sine 

Four  Place  Trigonometric  Functions 


271 


[Characteristics  of  Logarith 

ns  omitted  — determine  by  the  usual  rule  from  the  value] 

Radians 

Degrees 

Sine 

Tangent 

Cotangent 

Cosine 

Value 

Logio 

Value 

Logio 

Value 

Logio 

Value  Logio 

.4712 

27° 00' 

.4540 

.6570 

.5095 

.7072 

1.9626 

.2928 

.8910  .9499 

63°  00' 

1.0996 

.4741 

10 

.4566 

.6595 

.5132 

.7103 

1.9486 

.2897 

.8897  .9492 

50 

1.0966 

.4771 

20 

.4592 

.6620 

.5169 

.7134 

1.9347 

.2866 

.8884  .9486 

40 

1.0937 

.4800 

30 

.4617 

.6644 

.5206 

.7165 

1.9210 

.2835 

.8870  .9479 

30 

1.0^)08 

.4829 

40 

.4643 

.66(58 

.5243 

.7196 

1.9074 

.2804 

.8857  .9473 

20 

1.0879 

.4858 

50 

.4669 

.6692 

.5280 

.7226 

1.8940 

.2774 

.8843  .9466 

10 

1.0850 

.4887 

28°  00' 

.4695 

.6716 

.5317 

.7257 

1.8807 

.2743 

.8829  .9459 

62°  00' 

1.0821 

.4916 

10 

.4720 

.6740 

.5354 

.7287 

1.8676 

.2713 

.8816  .9453 

50 

1.0792 

.4945 

20 

.4746 

.6763 

.5392 

.7317 

1.8546 

.2683 

.8802  .9446 

40 

1.07(53 

.4974 

30 

.4772 

.6787 

.5430 

.7348 

1.8418 

.2652 

.8788  .9439 

30 

1.0734 

.5003 

40 

.4797 

.6810 

.5467 

.7378 

1.8291 

.2622 

.8774  .9432 

20 

1.0705 

.5032 

50 

.4823 

.6833 

.5505 

.7408 

1.8165 

.2592 

.8760  .9425 

10 

1.0676 

.5061 

29°  00' 

.4848 

.6856 

.5543 

.7438 

1.8040 

.2562 

.8746  .9418 

61°  00' 

1.0647 

.5091- 

10 

.4874 

.6878 

.5581 

.7467 

1.7917 

.2533 

.8732  .9411 

50 

1.0617 

.5120 

20 

.4899 

.6901 

.5619 

.7497 

1.7796 

.2503 

.8718  .9404 

40 

1.0588 

.5149 

30 

.4924 

.6923 

.5658 

.7526 

1.7675 

.2474 

.8704  .9397 

30 

1.0559 

.5178 

40 

.4950 

.6946 

.5(596 

.7556 

1.7556 

.2444 

.8689  .9390 

20 

1.0530 

.5207 

50 

.4975 

.6968 

.5735 

.7585 

1.7437 

.2415 

.8675  .9383 

10 

1.0501 

.5236 

30°  00' 

.5000 

.6990 

.5774 

.7614 

1.7321 

.2386 

.8660  .9375 

60° 00' 

1.0472 

.5265 

'     20 

.5025 

.7012 

.5812 

.7644 

1.7205 

.2356 

.8646  .9:368 

50 

1.0443 

.5294 

.5050 

.7033 

.5851 

.7673 

1.7090 

.2327 

.8631  .9361 

40 

1.0414 

.5323 

30 

.5075 

.7055 

.5890 

.7701 

1.6977 

.2299 

.8616  .9353 

30 

1.0385 

.5352 

40 

.5100 

.7076 

.5930 

.7730 

1.6864 

.2270 

.8601  .9346 

20 

1.0356 

.5381 

50 

.5125 

.7097 

.5969 

.7759 

1.6753 

.2241 

.8587  .9338 

10 

1.0327 

.5411 

31°  00' 

.5150 

.7118 

.6009 

.7788 

1.6643 

.2212 

.8572  .9331 

59° 00' 

1.0297 

.5440 

10 

.5175 

.7139 

.6048 

.7816 

1.6534 

.2184 

.8557  .9323 

50 

1.0268 

.5469 

20 

.5200 

.7160 

.6088 

.7845 

1.6426 

.2155 

.8542  .9315 

40 

1.0239 

.5498 

30 

.5225 

.7181 

.6128 

.7873 

1.6319 

.2127 

.8526  .9308 

30 

1.0210 

.5527 

40 

.5250 

.7201 

.6168 

.7902 

1.6212 

.2098 

.8511  .9300 

20 

1.0181 

.5556 

50 

.5275 

.7222 

.6208 

.7930 

1.6107 

.2070 

.8496  .9292 

10 

1.0152 

.5585 

32°  00' 

.5299 

.7242 

.6249 

.7958 

1.6003 

.2042 

.8480  .9284 

58°  00' 

1.0123 

.5614 

10 

.5324 

.7262 

.6289 

.7986 

1.5900 

.2014 

.8465  .9276 

50 

1.0094 

.5643 

20 

.5348 

.7282 

.6330 

.8014 

1.5798 

.1986 

.8450  .9268 

40 

1.00(35 

.5672 

30 

.5373 

.7302 

.6371 

.8042 

1.5697 

.1958 

.8434  .9260 

30 

1.0036 

.5701 

40 

.5398 

.7322 

.6412 

.8070 

1.5597 

.1930 

.8418  .9252 

20 

1.0007 

.5730 

50 

.5422 

.7342 

.6453 

.8097 

1.5497 

.1903 

.8403  .9244 

10 

.9977 

.5760 

33°  00' 

.5446 

.7361 

.6494 

.8125 

1.5399 

.1875 

.8387  .9236 

57°  00' 

.9948 

.5789 

10 

..5471 

.7380 

.6536 

.8153 

1.5301 

.1847 

.8371  .9228 

50 

.9919 

.5818 

20 

.5495 

.7400 

.6577 

.8180 

1.5204 

.1820 

.8355  .9219 

40 

.98i)0 

.5847 

30 

..5519 

.7419 

.(^619 

.8208 

1.5108 

.1792 

.8339  .9211 

30 

.9861 

.587(5 

40 

.5544 

.7438 

.6(561 

.8235 

1.5013 

.1765 

.8323  .9203 

20 

.9832 

.5905 

50 

.5568 

.7457 

.6703 

.8263 

1.4919 

.1737 

.8307  .9194 

10 

.9803 

.5934 

34°  00' 

.5592 

.7476 

.6745 

.8290 

1.4826 

.1710 

.8290  .9186 

56°  00' 

.9774 

.5%3 

10 

.5616 

.7494 

.(5787 

.8317 

1.4733 

.1683 

.8274  .9177 

50 

.9745 

.5992 

20 

.5640 

.7513 

.6830 

.8344 

1.4641 

.1656 

.8258] .9169 

40 

.9716 

.6021 

30 

.5664 

.7531 

.6873 

.8371 

1.4550 

.1629 

.8241  .91(30 

30 

.9687 

.6050 

40 

.5688 

.7550 

.(3916 

.8398 

1.4460 

.1602 

.8225  .9151 

20 

.9(557 

.6080 

50 

.5712 

.7568 

.6959 

.8425 

1.4370 

.1575 

.8208  .9142 

10 

.9628 

.6109 

35° 00' 

.5736 

.7586 

.7002 

.8452 

1.4281 

.1548 

.8192  .9134 

55°00' 

.9599 

.6138 

10 

.57(50 

.7604 

.7046 

.8479 

1.4193 

.1521 

.8175  .9125 

50 

.9570 

.6167 

20 

.5783 

.7622 

.7089 

.8506 

1.4106 

.1494 

.8158  .9116 

40 

.9541 

.6196 

30 

.5807 

.7640 

.7133 

.8533 

1.4019 

.1467 

.8141  .9107 

30 

.9512 

.6225 

40 

.5831 

.7657 

.7177 

.85.59 

1.3934 

.1441 

.8124  .9098 

20 

.9483 

.6254 

50 

.5854 

.7675 

.7221 

.8586 

1.3848 

.1414 

.8107  .9089 

10 

.9454 

.6283 

36°  00' 

.5878 

.7(i<)2 

.7265 

.8613 

1.3764 

.1387 

.8090  .9080 

54°  00' 

.9425 

Value 

Logio 

Value 

Login 

Value 

Logio 

Value  Logio 

Degrees 

Uadians 

Cosine 

Cotangent 

Tang 

ENT 

Sine 

272  Four  Place  Trigonometric  Functions 

[Characteristics  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  valae] 


Radians 

Degbees 

Sine 

Tangent 

Cotangent 

Cosine  - 

Value  Logio 

Value   Logio 

Value   Logic 

Value  Logic 

.6283 

36^^00' 

.5878  .7692 

.7265  .8613 

1.3764  .1387 

.8090  .9080 

64°  00' 

.^25 

.6312 

10 

.5901  .7710 

.7310  .8639 

1.3680  .1361 

.8073  .9070 

50 

.9396 

.6341 

20 

.5925  .7727 

.7355  .8666 

1.3597  .13M 

.8056  .9061 

40 

.9367 

.6370 

30 

.5948  .7744 

.7400  .8692 

l.;i514  .1308 

.8039  .9052 

30 

.9.338 

.6400 

40 

.5972  .7761 

.7445  .8718 

1.3432  .1282 

.8021  .9042 

20 

.9308 

.6429 

50 

.5995  .7778 

.7490  .8745 

1.3351  .1255 

.8004  .9033 

10 

.i>279 

.6458 

87° 00' 

.6018  .7795 

.7536  .8771 

1.3270  .1229 

.7986  .9023 

68°  00' 

.9250 

.6487 

10 

.6041  .7811 

.7581  .8797 

1.3190  .1203 

.7969  .9014 

50 

.9221 

.6516 

20 

.(i065  .7828 

.7627  .8824 

1.3111  .1176 

.7951  .9004 

40 

.9192 

.6545 

30 

.6088  .7»i4 

.7673  .8850 

1.3032  .1150 

.79;i4  .8995 

30 

.9163 

.6574 

40 

.6111  .7861 

.7720  .8876 

1.2954  .1124 

.7916  .8985 

20 

.9134 

.6603 

50 

.6134  .7877 

.7766  .8902 

1.2876  .1098 

.7898  .8975 

10 

.9105 

.6632 

88°  00' 

.6157  .7893 

.7813  .8928 

1.2799  .1072 

.7880  .8965 

62°  00' 

.9076 

.6661 

10 

.6180  .7910 

.7860  .89.54 

1.2723  .1046 

.7862  .8955 

50 

.9047 

.6(>90 

20 

.6202  .7926 

.7907  .8980 

1.2W7  .1020 

.7844  .8945 

40 

.9018 

.6720 

30 

.6225  .7911 

.7954  .9006 

1.2572  .0994 

.7826  .8935 

30 

.8988 

.6749 

40 

.6248  .7957 

.8002  .9032 

1.2497  .0968 

.7808  .8925 

20 

.8959 

.6778 

50 

.6271  .7973 

.8050  .9058 

1.2423  .0942 

.7790  .8915 

10 

.8930 

.6807 

89°  00' 

.6293  .7989 

.8098  .9084 

1.2349  .0916 

.7771  .8905 

61°00' 

.8901 

.6836 

10 

.6316  .8004 

.8146  .9110 

1.2276  .0890 

.7753  .8895 

50 

.8872 

.6865 

20 

.6338  .8020 

.8195  .9135 

1.2203  .0865 

.7735  .88^ 

40 

.8843 

.689i 

30 

.6361  .8035 

.8243  .9161 

1.2131  .0839 

.7716  .8874 

30 

.8814 

.6923 

40 

.6383  .8050 

.8292  .9187 

1.2059  .0813 

.7698  .8864 

20 

.878^) 

.6952 

50 

.6406  .806(> 

.8342  .9212 

1.1988  .0788 

.7679  .8853 

10 

.8756 

.6981 

40°  00' 

.6428  .8081 

.8391  .9238 

1.1918  .0762 

.7660  .8843 

60°  00' 

.8727 

.7010 

10 

.6450  .8096 

.8441  .9264 

1.1847  .0736 

.7642  .8832 

50 

.8698 

.7039 

20 

.6472  .8111 

.8491  .9289 

1.1778  .0711 

.7623  .8821 

40 

.86()8 

.7069 

30 

.(>494  .8125 

.8541  .9315 

1.1708  .0685 

.7604  .8810 

30 

.8639 

.7098 

40 

.(i517  .8140 

.8591  .9341 

1.1640  .0659 

.7585  .8800 

20 

.8610 

.7127 

50 

.6539  .8155 

.8642  .9366 

1.1571  .06U 

.7566  .8789 

10 

.8581 

.7156 

41°  00' 

.6561  .8169 

.8693  .9392 

1.1504  .0608 

.7547  .8778 

49°  00' 

.85,52 

.7185 

10 

.6583  .8184 

.8744  .9417 

1.1436  .0583 

.7528  .8767 

50 

.8523 

.7214 

20 

.6604  .811)8 

.87i)6  .9443 

1.1369  .0557 

.7509  .8756 

40 

.8494 

.7243 

30 

.6626  .8213 

.8847  .9468 

1.1303  .a532 

.7490  .8745 

30 

.8465 

.7272 

40 

.6648  .8227 

.8899  .9494 

1.1237  .0506 

.7470  .8733 

20 

.8436 

.7301 

50 

.6670  .8241 

.8952  .9519 

1.1171  .0481 

.7451  .8722 

10 

.8407 

.7330 

42°  00' 

.6691  .82.55 

.iXXH  .9544 

1.1106  .0456 

.7431  .8711 

48°  00' 

.8378 

.7359 

10 

.6713  .8269 

.9057  .9570 

1.1041  .0430 

.7412  .8699 

50 

.8*48 

.7389 

20 

.6734  .8283 

.9110  .9595 

1.0977  .0405 

.7392  .8G8S 

40 

.8319 

.7418 

30 

.6756  .8297 

.91(>3  .9()21 

1.0913  .0379 

.7373  .8676 

30 

.8290 

.7447 

40 

.6777  .8311 

.9217  .9646 

1.0850  .0354 

.7353  .8665 

20 

.8261 

.7476 

50 

.6799  .8324 

.9271  .9671 

1.0786  .0329 

.7333  .8653 

10 

.8232 

.7505 

48°  00' 

.6820  .8338 

.9325  .9697 

1.0724  .0303 

.7314  .8641 

47°  00' 

.8203 

.7534 

10 

.6841  .8351 

.9380  .9722 

1.0661  .0278 

.72<H  .8629 

50 

.8174 

.7563 

20 

.6862  .8365 

.9435  .9747 

1.0599  .0253 

.7274  .8618 

40 

.8145 

.7692 

30 

.6884  .8378 

.9490  .9772 

1.0538  .0228 

.7254  .8606 

30 

.8116 

.7621 

40 

.6905  .8391 

.9545  .9798 

1.0477  .0202 

.7234  .8594 

20 

.8087 

.7650 

50 

.6926  .8405 

.9601  .9823 

1.0416  .0177 

.7214  .8582 

10 

.8058 

.7679 

44°  00' 

.6947  .8418 

.9657  .9848 

1.0355  .0152 

.7193  .85<)9 

46°  00' 

.8029 

.7709 

10 

.6967  .8431 

.9713  .9874 

1.0295  .0126 

.7173  .8557 

50 

.7999 

.7738 

20 

.6988  .8444 

.9770  .9899 

1.0235  .0101 

.7153  .8545 

40 

.7970 

.7767 

30 

.7009  .»457 

.9827  .9924 

1.0176  .0076 

.7133  .8532 

30 

.7i^l 

.7796 

40 

.7030  .8469 

.9884  .9949 

1.0117  .0051 

.7112  .8520 

20 

.7912 

.7825 

50 

.7050  .i^82 

.9942  .9975 

1.0058  .0025 

.7092  .8507 

10 

.7883 

.7854 

46^00' 

.7071  M95 

1.0000  .0000 

1.0000  .0000 

.7071  .^495 

45°  00' 

.7854 

Value  Logjo 

Value   Login 

Value   Logio 

Value  Logio 

Degbees 

Radians 

Cosine 

Cotangent 

Tangent 

Shtf. 

ANSWERS 

[Answers  which  might  lessen  the  value  of  the  Exercise  are  not  given.] 
Pages  9-10.     5.    2|  miles.        16.    173.9  ft. 
Pages  12-13.     3.   22.        4.   ^(_bc  +  ca  +  ab) . 

Pages  16-17.     4.   i  rir2  sin  (02  -  0i)- 
5.    ^lroj'3  sin  ((^3  -  02)  +  nn  sin  (0i  -  03)+  nr2  sin  (02  -  0i)]. 

Page  21.    16.  They  intersect  at  [K»^i  +  a:2+a^3+X4),i(2/i  +  y2+2/3+2/4)]. 
19.    [Ka^i  +  a;2  +  a^a),  K^i  +  2/2  +  ^3)]- 

Pages  40-41.     6.   640/39.  9.    (&1W2  -  62^1)72  wima  (mi  -  W2). 

10.    (3,1). 

Pages  46-47.    2.   (a)  r  sin  0  =  ±  5  ;  (&)  r  cos  0  =  ±  4 ; 
(c)  rcos(0--f7r)  =  ±  12. 

3.  0  =  0,  r  sin  0  =  9,  0  =  i  TT,  r  cos  0  =  6.        14.   8464/85. 

19.    (_  5,  _  10).        21.   x  =  l  (by  inspection),  4x-3?/  +  16  =  0. 

Page  49.    4.   h"^  ^  ah^O. 

Page  50.     1.   tan-i  ^^^^'"^^ ;  a  =-  6,  /i^  =  a6. 

a  +  & 

4.  [wi(&2  -  &)-  ^2(61  -  6)]V2mim2  (W2-  wii). 

6.  r(2cos  0  —  3sin0)+ 12  =0. 

10.   1  hr.  10  m. ;  176  miles  from  Detroit. 

Pages  54-55.     6.  x^  +  ^/^  -  96  x  -  54  y  +  2408  =  0  ;  31 .8  ft.  or  66.3  ft. 
8.   x^  +  1/2  —  16 X  +  8 ?/  +  60  =  0.  9.   A  circle  except  for  k  =±  1. 

10.  x^  +  y'^  +  4:l±^x  +  4  =  0. 

Page  56.    2.    (a)  r2  -  20  r  sin  0  +  76  =  0; 

(5)  r2- 12rcos(0-i7r)+18  =  O;  (c)  r  +  8  sin  0  =  0. 

Page  58.    3.    (-  6,  -  1),  (29/106,  42/53). 

7.  8x-4?/-ll±15V2  =  0. 

273 


274  ANSWERS 

Page  60.    3.    (xi  -  /t) (x  -  A)  +  (yi  -k)(y  -k)=  r^. 

7.  (-rM/C,  -r2J5/C).        8.    (2,1). 

Page  62.     6.    (x  -  79/38)2  +  (y  -  55/38)2  =  (65/38)2. 

8.  x2  +  y2  +  4  X  -  2  y  -  15  =  0. 

Page  67.     1.    (c)  Polar  lies  at  infinity. 

Pages  69-70.     3.    Let  Z,  M  be  the  intersections  of  the  circle  with 
OPi,  then  d^  -  r2  =  LPi  •  MPi. 
4.  x  =  yi  Vi(a  +  6)2 -4c. 

6.  (c)  2x2+2y24-22x  +  6y  +  15  =  0,  2x2+2y2_i0x-10y-25  =  0. 

9.  6x2  ^  5^2  ^  fj2^x  —  a'^y  =  0. 

12.  If  the  vertices  of  the  square  are  (0,  0),  (a,  0),  (0,  a),  (a,  a)  and  k^ 
is  the  constant,  the  locus  is  2  x2  +  2  y2  _  2  ax  —  2  ay  +  2  a^  —  A;2  =  0  ; 
k>a;  ^aV6. 

13.  If  the  vertices  of  the  triangle  are  (a,  0),  (—a,  0),  (0,  aV3)  and 
A;2  is  the  constant,  the  locus  is  3  x2  +  3  y2  _  2  VS  ay  +  3  a2  —  2  A;2  =  0. 

Pages  74-75.  10.  (a)  2y=3x2  +  5x;  (6)  12y  =- 5x2  + 29x  -  18. 
11.  300y  =-x2  + 230x;  44.1  ft.  above  the  ground;  230  ft.  from  the 
starting  point. 

Page  81.  6.  East,  East  33°  41'  North,  East  63°  8'  North,  East  18°  26' 
South. 

10.  100/(7r  +  4). 

Pages  84-85.     10.   0,  8°  8'.        11.   7°  29'. 
15.    When  the  side  of  the  square  is  3  in. 
17.    (a)  6y  =  x8  +  6x2-19x;  (6)  7y  =  2x«  -  x2  -  29a;  +  35. 

Page  92.     10.     -  1.88,  1.53,  .347. 

Pages  97-98.  2.  (a)  (4,  jir),  (4,  |t);  (6)  (a,  ^tt),  (a,  |  tt)  ; 
(c)  (4,0);  (d)  (4  a,  JT),  (4a,  fx). 

7.  (a)  y2_4a;_|-4  =  0;  (6)  14y2_  45  a;  +  62y  +  60  =  0. 

8.  (6)  x2-10x-3y  +  21=0;  (c)  a;2  +  2x  +  y  -  1  =0. 

9.  The  equation  of  a  parabola  contains  an  xy  term  when  its  axis  is  oblique 
to  a  coordinate  axis. 

Pages  106-108.     8.    (a)  y  =  0  ;  (6)  2x  +  2y  -  9=0,  2  x-y-18  =  0; 
(c)  2x  +  2y-9  =  0,  8x+  16y-27  =0,  24x-16y-  153  =  0. 
id)  8x-16y-27  =  0. 

14.  y  =  kx.        16.   Directrix;  y2  =  a(x  -  3 a).       21.    —  (1  +  w2). 

*n.2 


ANSWERS  275 

26.   x2-80x-2400?/  =  0;  0,  -  i,  -  |,  -  J,  0,  |,  2. 
29.   x2  =  360(y-20). 

3 

Page  115.    2.    (3  7r-4)/6  7r.        3.    §a^(^+f')^. 

3  m** 

6.   (a)  64/3  ;  (6)  625/12  ;  (c)  1/12.         7.    123.84  ft^.         8.    1794 J  tons. 

9.    199.4  ft2. 

Page  118.    9.   8x2 -2  a;?/ +  8  2/2 -63  =  0. 

Page  122.     10.    3x2  -  ^2  =  3  ^^^        n.    5.         14.   2xy  =  1. 

Pages  128-129.    2.  ^X+-r=:c2.     13.  54.5  ft.,  42.2  ft.     17.62/^2. 
X  y 

20.   An  ellipse  or  hyperbola  according  as  one  circle  lies  within  or  without 

the  other  circle. 

Pages  138-139.     7.  (a)  A^a^-&h'^=^  C^  ;  (&)  aP-  cos2 ^- &2 sin2 ^=p2. 
19.    62.  21.    a2-f  62;  a2_52.  22.   4  a6.  23.   sin-i  (a6/a'6')- 

25.    (a)  x2  +  y2  ^  Qj2  +  52  .  (6)  x2  +  ^2  ^  (j2  _  52. 

Page  144.     3.   (a)  (1,  -  1),  (1  ±  V2,  -  1),  x  =  1  ±  |  V2  ; 
(&)    a,0),  (f,0),  (-f,0),X  =  0,X=:l. 

4.  2  62/a.     7.   (a)  a2i/2  =  h'h:,{a  -  x);  (6)  52^2  =  cfiyQy  _  yy 
8.   Two  straight  lines. 

Page  151.     2.    (a);  Vertices  (5,  3),  (8,  3);  semi-axes  3/2,  V2. 
(6)  Vertices  (4,  8/3),  (8,  8)  ;  semi-axes  10/3.  5\/3/3. 

(c)  Vertices  (17/5,  7/5),  (1,  3) ;  semi-axes  V65/5,  ^13/2. 
3.   3x +2y- 2  =  0;  21/13,  -37/26,  10/ \/l3. 

Page  153.     5.    (acos  ^,  —  asin  ^),  x2 -|- y2  _  2  a(xcos0  —  ?/ sin  ^)=  0. 

Pages  161-162.    2.  (a)  3 x  -  14 1/  =  0 ;  (6)  y  =-  3/13,  x  =  -  14/13. 

5.  2  x2  -  x?/  -  15  ?/2  H-  X  -f-  19  ?/  -  6  =  0, 
2  x2  -  xy  -  15  ?/2  +  x  -f  19  ?/  -  28  =  0. 

6.  6x2 -f  xy- 2  2/2 -9x  4-82/ -46  =  0, 
6x^  +  xy-2y^  -9x  +  8y  +  Si  =0. 

11.    (a)  x2/4  +  ?/2  =  1 ;  (b)  x2/4  -  2/2/2  =  1  ;  (c)  3x2  -f  y2  +  6  =  0; 

(d)  x2/16  +  2/2/4  =  1  .     (e)   (3  -h  Vl7)x2  -f  (3  -  V17)?/2  =  4  ; 
(/)   (2-hV2)x2-H(2-V2)2/2  =  l. 

15.   x^-\-y^=  aK 

19.   Equilateral  hyperbola. 


276 


ANSWERS 


Page  168.     2.    (a)  Simple  point ;     (6)  node  ;     (c)  cusp  ;     (d)  cusp. 
4.    (a)  None  ;  (b)  node  at  {b,  0)  ;  (c)  isolated  point  at  (a,  0)  ; 

(d)  cusp  at  (a,  0). 

Pages  174-175.     4.    r=a(sec  0  ±  tan  0)  or  (x  -  a)y^-{-  x'^(x-^  a)  =0. 
10.  a;2?/2  =  a2(a;2  ^  yiy  ^    Cissoid  (a  -  x)y^  =  x^. 

12.    y(x2  4-  y2)  :=  rt(^2  _  ^2).  i3_    y  _  ^  ctn  <p. 


l  +  V 


etc. 


Page  195.     6.  — :^ 

V2(l  +  ;Z'  +  mm'  +  nn') 
13.    \{xi  +  Xa  +  xa),  K^i  +  ^2  +  ys),  i(;?i  +  22  +  ^s). 

Page  199.     6.   cos-i(7/3V29). 

Page  203.     2.    ^V465.        3.    fV269. 
6.   (3962,  AV  43',  276°  16'),  (320,  -  2914,  2666),  2931. 


7.    \  rir2  Vl  —  [cos  ^1  cos  ^2  +  sin  ^1  sin  62  cos  (0i  —  02)]*- 


8.    Vri^  4.  ra'-^  —  2  rir2[sin  ^1  sin  62  cos  (0i  —  02)  +  cos  di  cos  ^2]. 
10.    -  1,  10,  7. 

Page  208.     3.   30a:- lOy  +  7^  -  89  =  0. 

6.    97/28,  -  97/49,  -  97/9.  7.   3x-4y  +  2«-6  =  0. 

Page  212.     5.    4x  +  8y-f0  =  81,4x+8y+«  =  9O. 
Page  215.     2.    (a)  56/3 ;  (6)  0 ;  (c)  19/3. 

12.    3x-2y  =  l.  13.    6x  +  lly  +  9;?  =  58. 


Page  218 

16.    70°  31'. 


17.    cos-i(2  K^  +  3  a2)/(4  K^  +  3  a-^). 


Pages  226-228.     3.    69°  29'. 
21.   X  -  2  y  +  «  +  8  =  0. 

y^i  —  xi  y2  —  yi  Z2  —  21 
24.         a\  b\  cx 

a^  62  C2 


19.    (a)    V63719;  (6)   V'194/33. 


Pages  236-237.     4. 

7.   x2-  3?/2_3«'^  =  0. 


(1,  0,  -  3),  (-  9/11,  20/11,  27/11). 
13.    25  (x-^  +  y2  +  5r2)  =  162^  25  «  =  64. 


Pages  240-241.     4.    (4,-5,-3).        6.    (4,6,2). 
6.    bx-\-2y-  z  =  2b,  2x-3y  +  2  +  25  =  0. 

Page  246.     4.    (x^  +  j,2  +  ^^2  _  ^2  _  52)2  _  4  ^,2(^2  _  ^2)  =  q. 

6.  (a)  16a2(x2  +  2:2^  =  y4.  (5)  I6a2[(x-a)2+22]  =  (4a-2_y2)2. 

7.  y2(a;2  +  2-2)  =  a*. 


INDEX 


(The  numbers  refer  to  the  pages.) 


Abscissa,  1,  4. 

Acnode,  167. 

Adiabatic  expansion,  188. 

Algebraic  curves,  163-168. 

Amplitude,  15. 

Anchor-ring,  246. 

Angle  between  line  and  plane,  224 ; 

between  two  curves,  84 ;    between 

two  lines,  39,   196,  223;    between 

two  planes,  211. 
Anomaly,  15. 
Area   of   ellipse,    138 ;     of   parabolic 

segment,  112-115;   of  triangle,  11, 

12,  200;   under  any  curve,  114. 
Asymptotes,  121. 
Asymptotic  cone,  255. 
Axes    of    coordinates,    4,     189 ;     of 

ellipse,  116;   of  hyperbola,  120. 
Axis,    17;    of   parabola,    72,   94;    of 

pencil,    215 ;     of   revolution,    244 ; 

of  symmetry,  76. 
Azimuth,  15. 

Bisecting  planes,  211. 

Bisectors  of  angles  of  two  lines,  45. 

Boyle's  Law,  161. 

Cardioid,  170. 

Cartesian  coordinates,  16. 

Cartesian  equation  of  conic,  142 ;  of 
ellipse,  117;  of  hyperbola,  120; 
of  parabola,  95. 

Cartesius,  16. 

Cassinian  ovals,  171,  174,  262. 

Catenary,  108. 

Center  of  ellipse,  116,  132;  of  ellip- 
soid, 243;  of  hyperbola,  120,  132; 
of  hyperboloid,  248,  249  ;  of  inver- 
sion, 63;  of  pencil,  48;  of  sheaf, 
216;    of  symmetry,  76,  132. 


Centroid,  21. 

Chord  of  contact,  65. 

Circle,  51-70  ;  in  space,  232  ;  through 

three  points,  61. 
Circular  cone,  251. 
Cissoid,  170. 
Classification     of     conies,     142 ;      of 

quadric  surfaces,  252-255. 
Clockwise,  11. 
Colatitude,  202. 
Common  chord,  69. 
Completing  the  square,  52,  73. 
Component,  18,  192,  196. 
Conchoid,  169. 
Cone,  251,  261 ;   asymptotic,  255;  of 

revolution,  252. 
Conic  sections,  140-147,  148. 
Conies  as  sections  of  a  cone,  144-147. 
Conjugate    axes,     238 ;      axis,     121 ; 

diameters,    132-136 ;     hyperbolas, 

122  ;  lines,  238. 
Continuity,  90-91, 
Contour  lines,  261. 
Coordinate  axes,  4,  189  ;  planes,  189  ; 

trihedral,  189. 
Coordinates,    1,    5,    189;    polar,    15, 

202. 
Coplanar,  198. 
Cosine  curve,  176. 
Counterclockwise,  11. 
Cross-sections,  243,  247,  249,  260. 
Crunode,  167. 

Cubic  curves,  163  ;    function,  82-84. 
Curve  in  space,  205. 
Cusp,  167. 
Cycloid,  172. 
Cylinders,  261. 


Derivative,    79-81,    82,    87-89;     of 
ax^,  79 ;    of  cubic  function,  82 ;    of 


277 


278 


INDEX 


polynomial,    87-89 ;    of   quadratic 

function,  80 ;  of  z",  89. 
Derived  curves,  92. 
Descartes,  16. 

Determinant,   11,  12;    of  two  equa- 
tions, 37. 
Diameter,   243;    of  ellipse,   132;    of 

hyperbola,  135  ;   of  parabola,  105- 

106. 
Direction  cosines,  194,  219. 
Director  circle,  139  ;   sphere,  259. 
Directrices  of  conies,  140,  143. 
Directrix  of  parabola,  93. 
Discriminant  of  equation  of  second 

degree,     156-157 ;      of     quadratic 

equation,  56. 
Distance  between  two  points,  7,  16, 

190;    of  point  from  line,  44,  225; 

of  point  from  origin,   6,   190 ;    of 

point    from    plane,    210;     of    two 

lines,  225-226. 
Division  ratio,  3,  8,  193. 
Double  point,  166-167. 

Eccentric  angle,  137. 
Eccentricity,  125,  140. 
Ellipse,  116-139,  140,  146,  157-159. 
Ellipsoid,    242-244;     of    revolution, 

244. 
Elliptic  cone,  251 ;   paraboloid,  250. 
Empirical  equations,  178-188. 
Epicycloid,  173. 
Equation  of  first  degree,  see  Linear 

equation;     of    line,    25,    31;     of 

plane,  205-209  ;    of  second  degree, 

52. 
Equations  of  line,  219-220. 
Equator,  244. 
Equatorial  plane,  202. 
Equilateral  hyperbola,  121. 
Euler's  angles,  265. 
Exponential  curve,  177. 

Factor  of  proportionality,  24. 
Falling  body,  30,  50,  74. 
Family  of  circles,  69  ;  of  spheres,  240. 
Foci  of  conic,    143;    of  ellipse,   116, 

140;   of  hyperbola,  119,  140. 
Focus  of  parabola,  93. 
Four-cusped  hypocycloid,  174. 


Function,  28;  linear,  71;  of  two 
variables,  261 ;  quadratic,  71 ; 
rational  integral,  85. 

Gas-meter,  26,  181. 

Gas  pressure,  184,  188. 

General  equation  of  second  degree, 

52,  149-162,  229,  252. 
Geometric    properties    of    parabola, 

104. 

Higher  plane  curves,  163-188. 
Hooke's  Law,  14,  24,  29,  179,  181. 
Hyperbola,  119-139,  140,   146,   157- 

159. 
Hyperbolic  paraboloid,  250;    spiral, 

174. 
Hyperboloid,  of  one  sheet,  247-248; 

of  revolution  of  one  sheet,  248 ;   of 

revolution  of  two  sheets,  249 ;    of 

two  sheets,  248-249. 
Hypocycloid,  174. 

Imaginary  ellipsoid,  249. 

Inclined  plane,  183. 

Inflection,  83. 

Intercept,  25,  33 ;   form,  32,  207. 

Intersecting  lines,  219. 

Intersection  of  line  and  circle,   56 ; 

of  line  and  ellipse,  130 ;  of  line  and 

parabola,  102  ;    of  line  and  sphere, 

234  ;  of  two  lines,  37. 
Invariants,  162. 
Inverse  of  a  circle,  63  ;  trigonometric 

curves,  176. 
Inversion,  62,  236. 
Inversor,  111. 
Isolated  i)oint,  167. 

Latitude,  202. 

Latus  rectum  of  parabola,  94 ;  of 
conic,  141. 

Left-handed  trihedral,  264. 

Lemniscate,  172,  174. 

Level  lines,  261. 

LimaQon,  169. 

Limiting  cases  of  conies,  147. 

Line,  23,  219  ;  and  plane  perpendicu- 
lar at  given  point,  224 ;  of  nodes, 
265;    intercept  form,  32;    normal 


INDEX 


279 


form,  42 ;  parallel  to  an  axis,  22 ; 
parameter  equations,  220  ;  through 
one  point,  34,  220  ;  through  origin, 
23  ;   through  two  points,  35,  220. 

Linear  equation,  31,  205. 

Linear  equations,  three,  214 ;  two, 
37-38,  205,  214. 

Linear  function,  28,  71. 

Lituus,  174. 

Logarithmic  curve,  177. 

Logarithmic  paper,  186;  plotting, 
184-188. 

Longitude,  202. 

Major  axis,  117. 

Maximum,  80,  82. 

Mechanical  construction  of  ellipse, 
116  ;  of  hyperbola,  119  ;  of  parab- 
ola, 95. 

Melting  point  of  alloy,  181. 

Meridian  plane,  202  ;   section,  245. 

Midpoint  of  segment,  9. 

Minimum,  80,  82. 

Minor  axis,  117. 

Moment  of  a  force,  200-201. 

Multiple  points,  168. 

Nodal  line,  265. 

Node,  167. 

Non-linear     equations     representing 

lines,  49,  216. 
Normal  form,  42,  208. 
Normal  to  ellipse,  126 ;   to  parabola, 

101,  103 ;   to  any  surface,  259. 

Oblate,  244. 
Oblique  axes,  6,  7,  190. 
Octant,  189. 
Ordinary  point,  166. 
Ordinate,  5. 
Origin,  1,  4,  189. 

Orthogonal  substitution,  262  ;  trans- 
formation, 262. 

Parabola,  71-81,  93-115,  145,  159- 
160 ;  Cartesian  equation,  95 ; 
polar  equation,  93-94  ;  referred  to 
diameter  and  tangent,  110. 

Paraboloid,  elliptic,  250 ;  hyperbolic, 
250:  of  revolution,  251. 


Parallel  circle,  245  ;   planes,  260. 
Parallelism,  27,  32,  40,  197,  223,  224. 
Parallelogram  law,  18. 
Parameter,    69,    109 ;     equations    of 

circle,    109;     of    ellipse,    137;     of 

hyperbola,   137;    of  line,  220;    of 

parabola,  110. 
Peaucellier's  cell.  111. 
Pencil  of  circles,  69  ;   of  lines,  48 ;   of 

parallels,  48 ;    of  planes,  215 ;    of 

spheres,  240. 
Pendulum,  74. 
Perpendicularity,    27,    32,    40,    197, 

223,  224. 
Plane,  204-218 ;  intercept  form,  207  ; 

normal  form,  208 ;    through  three 

points,  207. 
Plane  and  line  perpendicular  at  given 

point,  224. 
Plotting  by  points,  30,  71. 
Points  of  inflection,  83. 
Polar,  64,  66,  237;    angle,  15;    axis, 

15,    202;      coordinates,     15,    202; 

equation  of   circle,    55 ;     of   conic, 

141-142  ;   of  line,  41 ;   of  parabola, 

93-94 ;   plane,  237. 
Pole,  15,  202. 

Pole  and  polar,  64,  66,  237. 
Poles,  244. 

Polynomial,  85-92  ;   curve,  90-92. 
Power  of  a  point,  68,  239. 
Projectile,  75,  81,  111. 
Projecting  cylinders,  232 ;    planes  of 

a  line,  221-222. 
Projection,  17-20,  192-193,  196. 
Prolate,  244. 

Proportional  quantities,  23. 
Pulleys,  26,  30,  36,  180. 
Pythagorean  relation,  194. 

Quadrant,  5. 

Quadratic    equation,    56 ;     function, 

71-81. 
Quadric  surfaces,  242-261,  252. 

Radical   axis,    68,    239-240;    center, 

69,  239-240 ;  plane,  239. 
Radius  vector,  15,  194,  202. 
Rate  of  change,  28,  87 ;    of  interest, 

28,  34. 


280 


INDEX 


Reciprocal  polars,  238. 

Rectangular  coordinates,  6 ;  hyper- 
bola, 121. 

Reduction  to  normal  form,  43,  209. 

Related  quantities,  13. 

Removal  of  term  in  xy,  154. 

Resultant,  18,  192. 

Right-handed  trihedral,  264. 

Rotation  of  axes,  151-152 ;  of  coor- 
dinate trihedral,  262-265. 

Ruled  surfaces,  257-259. 

Rulings  on  hyperboloid  of  one  sheet, 
258  ;  on  hyperbolic  paraboloid,  259. 

Second  derivative,  83. 

Sheaf  of  planes,  216. 

Shortest  distance  of  two  lines,  225-226. 

Simple  point,  166. 

Simple  harmonic  motion,  177. 

Simpson's  rule,  114. 

Simultaneous  linear  equations,  37- 
38,  214. 

Sine  curve,  175. 

Slope,  23 ;  of  ellipse,  125 ;  of  hyper- 
bola, 128 ;  of  parabola,  78-79,  99  ; 
of  secant  of  parabola,  78. 

Slope  form  of  equation  of  line,  25. 

Sphere,  229-241 ;  through  four 
points,  231. 

Spherical  coordinates,  202. 

Spheroid,  244. 

Spinode,  167. 

Spiral  of  Archimedes,  174. 

Statistics,  13. 

Straight  line,  22. 

Strophoid,  174. 

Subnormal  to  parabola,  101. 

Substitutions,  182. 

Subtangent  to  parabola,  101. 

Superposable  axes,  152 ;  trihedrals, 
264. 

Surface,  204,  260 ;  of  revolution,  245. 


Suspension  bridge,  108. 
Symmetry,  76-77,  132. 

Tangent  to  algebraic  curve  at  origin, 
165-168 ;  to  circle,  59 ;  to  ellipse, 
124,  130 ;  to  hyperbola,  128,  130 ; 
to  parabola,  78,  100,  102. 

Tangent  cone  to  sphere,  235. 

Tangent  curve,  176. 

Tangent  plane  to  ellipsoid,  256-257; 
to  hyperboloids,  257 ;  to  parabo- 
loids, 257  ;  to  quadric  surfaces,  257  ; 
to  sphere,  233. 

Telescope,  109. 

Temperature,  14,  30,  182. 

Tetrahedron  volume,  213. 

Thermometer,  2,  30,  34. 

Torus,  246. 

Transcendental  curves,  176. 

Transformation  from  cartesian  to 
polar  coordinates,  16,  202-203; 
to  center,  142,  156 ;  to  parallel 
axes,  11,  155. 

Translation  of  axes,  11,  149-150;  of 
coordinate  trihedral,  199. 

Transverse  axis,  121. 

Trochoid,  173. 

Uniform  motion,  29,  50. 
Units,  5. 

Vector,  17,  192. 
Vectorial  angle,  15. 
Velocity,  29,  30. 
Versiera,  170. 
Vertex  of  parabola,  72,  94. 
Vertices  of  ellipse,   116;    of  hyper- 
bola, 120. 
Volume  of  tetrahedron,  213. 

Water  gauge,  2. 
Whispering  galleries,  129. 


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BY 

WALTER  BURTON   FORD 

Junior  Professor  of  Mathematics,  University  of  Michigan 

And  CHARLES  AMMERMAN 

The  William  McKinley  High  School,  St.  Louis 

Edited  by  Earle  Raymond  Hedrick,  Professor  of  Mathematics 

in  the  University  of  Missouri 

Plane  and  Solid  Geometry,  cloth,  i2mo,  31  q  pp.,  $1^5 
Plane  Geometry,  cloth,  i2mo,  213  pp.,  $  .80 
Solid  Geometry,  cloth,  i2mo,  106  pp.,  $  .80 

STRONG  POINTS 

I.  The  authors  and  the  editor  are  well  qualified  by  training  and  experi- 
ence to  prepare  a  textbook  on  Geometry. 

II.  As  treated  in  this  book,  geometry  functions  in  the  thought  of  the 
pupil.     It  means  something  because  its  practical  applications  are  shown. 

III.  The  logical  as  well  as  the  practical  side  of  the  subject  is  emphasized. 

IV.  The  arrangement  of  material  is  pedagogical. 

V.  Basal  theorems  are  printed  in  black-face  type. 

VI.  The  book  conforms  to  the  recommendations  of  the  National  Com- 
mittee on  the  Teaching  of  Geometry. 

VII.  Typography  and  binding  are  excellent.  The  latter  is  the  reenforced 
tape  binding  that  is  characteristic  of  Macmillan  textbooks. 

"  Geometry  is  likely  to  remain  primarily  a  cultural,  rather  than  an  informa- 
tion subject,"  say  the  authors  in  the  preface.  "  But  the  intimate  connection 
of  geometry  with  human  activities  is  evident  upon  every  hand,  and  constitutes 
fully  as  much  an  integral  part  of  the  subject  as  does  its  older  logical  and 
scholastic  aspect."  This  connection  with  human  activities,  this  application 
of  geometry  to  real  human  needs,  is  emphasized  in  a  great  variety  of  problems 
and  constructions,  so  that  theory  and  application  are  inseparably  connected 
throughout  the  book. 

These  illustrations  and  the  many  others  contained  in  the  book  will  be  seen 
to  cover  a  wider  range  than  is  usual,  even  in  books  that  emphasize  practical 
applications  to  a  questionable  extent.  This  results  in  a  better  appreciation 
of  the  significance  of  the  subject  on  the  part  of  the  student,  in  that  he  gains  a 
truer  conception  of  the  wide  scope  of  its  application. 

The  logical  as  well  as  the  practical  side  of  the  subject  is  emphasized. 

Definitions,  arrangement,  and  method  of  treatment  are  logical.  The  defi- 
nitions are  particularly  simple,  clear,  and  accurate.  The  traditional  manner 
of  presentation  in  a  logical  system  is  preserved,  with  due  regard  for  practical 
applications.     Proofs,  both  formal  and  informal,  are  strictly  logical. 


THE   MACMILLAN  COMPANY 

Publishers  64-66  Fifth  Avenue  New  York 


Elements  of  Theoretical  Mechanics 

BY 

ALEXANDER   ZIWET 

Professor  of  Mathematics  in  the  University  of  Michigan 

Cloth,  8vo,  4Q4  pp.,  $4.00 

The  work  is  nof  a  treatise  on  applied  mechanics,  the  applications  being 
merely  used  to  illustrate  the  general  principles  and  to  give  the  student  an  idea 
of  the  uses  to  which  mechanics  can  be  put.  It  is  intended  to  furnish  a  safe 
and  sufficient  basis,  on  the  one  hand,  for  the  more  advanced  study  of  the  sci- 
ence, on  the  other,  for  the  study  of  its  more  simple  applications.  The  book 
will  in  particular  stimulate  the  study  of  theoretical  mechanics  in  engineering 
schools. 


Introduction  to  Analytical  Mechanics 

BY 

ALEXANDER   ZIWET 

Professor  of  Mathematics  in  the  University  of  Michigan 

And  peter   FIELD 

Assistant  Professor  of  Mathematics  in  the  University  of 

Michigan 

Cloth,  i2mo,  374  pp.,  $1.60 

The  present  volume  is  intended  as  a  brief  introduction  to  mechanics  for 
junior  and  senior  students  in  colleges  and  universities.  It  is  based  to  a  large 
extent  on  Ziwet's  "Theoretical  Mechanics  ";  but  the  applications  to  engineer- 
ing are  omitted,  and  the  analytical  treatment  has  been  broadened.  No  knowl- 
edge of  differential  equations  is  presupposed,  the  treatment  of  the  occurring 
equations  being  fully  explained.  It  is  believed  that  the  book  can  readily  be 
covered  in  a  three-hour  course  extending  throughout  a  year.  The  book  has, 
however,  been  arranged  so  that  certain  omissions  may  be  easily  made  in  order 
to  adapt  it  for  use  in  a  shorter  course. 

While  more  prominence  has  been  given  to  the  analytical  side  of  the  sub- 
ject, the  more  intuitive  geometrical  ideas  are  generally  made  to  precede  the 
analysis.  In  doing  this  the  idea  of  the  vector  is  freely  used;  but  it  has 
seemed  best  to  avoid  the  special  methods  and  notations  of  vector  analysis. 

That  material  has  been  selected  which  will  be  not  only  useful  to  the  begin- 
ning student  of  mathematics  and  physical  science,  but  which  will  also  give  the 
reader  a  general  view  of  the  science  of  mechanics  as  a  whole  and  afford  him 
a  foundation  broad  enough  to  faciUtate  further  study. 


THE   MACMILLAN   COMPANY 

Publishers  64-66  Fifth  Avenue  New  Tork 


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